\section{Introduction}\label{INTRODUCTION}

Ann decides to go to her local Italian restaurant for dinner. Ann whittles down her choice to either the spaghetti or the risotto. The spaghetti is always the same and always pretty good - a safe choice. The risotto is risky, sometimes it is excellent but on occasion it has been bad. Ann suffers from regret. If she chooses the risotto and it turns out to be bad then she will regret her choice; likewise, if she chooses the spaghetti and the risotto turns out to be be excellent. If Ann is regret averse, then she will account for the possibility of these unpleasant regretful experiences when making her decision.

Information plays an integral role in the above story. Ann's regret is evaluated as the utility loss experienced from comparing a choice made - that turned out to be suboptimal - to a foregone alternative. But implicit in this construction is that such a comparison can always be made. While this may be true for financial assets listed on an exchange, there are many situations in life where information about unchosen options is not automatically available, as in the case of our restaurant example as it is uncommon to order two main courses. In this paper we allow for this possibility, by allowing for the ex-post information available to a decision maker to depend on choice.

Now suppose the environment is modified slightly in that if Ann orders the spaghetti then she will never learn the quality of the risotto, but since the spaghetti is always the same its quality is learned whether it is chosen or not. In this environment there is an informational asymmetry. Does this matter? The answer is:\ yes, it can matter because differing levels of ex-post information matter for behavioural individuals like Ann who factor in ex-post information into their decision-making. In this new environment, the spaghetti dish is now more relatively desirable than before because choosing it completely insures against regret, while the total benefit associated with choosing the risotto remains unchanged. Depending on the parameters, it is easy to see that ex-post information structures varying with choice can lead to preference reversals.

People may learn from others. Imagine that Ann has a friend called Barry who joins her for dinner. If Barry orders the risotto, then ordering the spaghetti is no longer a safe haven from regret for Ann as it was when she was dining alone. Previously by ordering the spaghetti, Ann was completely insured against regret. But now, if Ann orders spaghetti and learns via Barry's that the risotto was excellent, then she will experience regret. In this case, Barry's choice has imposed an externality on Ann, because his ordering the risotto Barry will impact Ann's ex-post information, which in turn impacts the psychological payoff (i.e., the payoff from the choice made plus the potential psychological loss due to regret) she associates with a given choice. So, what on the surface appeared to be two independent decision problems made in isolation is in fact a behavioural game since each person's choice can reveal information to others.

In this paper we present a model that captures the Ann and Barry story above.\footnote{The example is based on \citet{ArielyLevav:2000:JCR}. In that paper, the focus was on the differences in patrons' orders in two different settings:\ one where patrons order simultaneously and another where they order sequentially. \citet{ArielyLevav:2000:JCR} noted disparities in order choice (i.e., the propensity to ``coordinate'') depending on the protocol. The model in this paper can explain this finding.} While the model can handle environments far richer than this example, we believe that the simple story captures all the salient features. That is, we start out with the (obvious) observation that ex-post information matters for regret. We then provide some formal machinery that allows the analyst to model how the ex-post information available to a regret averse individual can vary with choice and hence affect optimal choice.\footnote{In fact we allow for the possibility that the ex-post information structure is not fixed. That is, there may be multiple ex-post informational environments associated with a given choice and there is a given probability distribution over the different ex-post information structures.} Effectively, this requires extending the domain of preferences from `objects of choice' to `objects of choice and their associated information environment'. Our first result, Theorem \ref{thm:regretEquivalent}, shows that when there is full information ex-post information, an individual with standard preferences and an individual who is regret averse with a linear regret term are observationally equivalent in their choice behaviour.

We then observe how Theorem \ref{thm:regretEquivalent} breaks down once ex-post information depends on choice. In our opinion it seems important that experimenters are aware of this fact.  A full classification of precisely how and when ex-post information environments impact optimal choice is not possible: not all information sets can be ordered. But our second result, Theorem \ref{thm:lessInfo}, shows how information sets can be partially ordered. In particular, we show that `more' ex-post information (formally defined later) is never desirable for a regret averse decision maker. Ignorance is bliss. In a sense this is obvious:\ a regret averse individual `zeroes-in' on the best-performing lottery that is learned about in each state, creating a state-dependent reference point that can only be matched but never exceeded. So when the number of outcomes that are learned about increases, regret can only go up.

After this, we allow for the possibility that the ex-post information available depends not only on one's own choice but also on the choices of others. Precisely, we consider an environment wherein a large number of regret averse individuals choose from a common choice set. We show that such an environment is not a series of independent decision problems to be analysed in isolation, but is in fact a rich multi-player behavioural game, that we term the regret game. When the behaviour of others can impact one's ex-post information, for certain parameter specifications the regret game is a game of coordination that admits multiple equilibria; this despite the fact that the same individuals all have a strictly dominant choice when faced with the same decision problem in isolation. Theorem \ref{thm:Nashmany} is a precise statement of this.

The term `asymmetric information' is typically understood to mean `ex-ante asymmetric information'. The idea is that in advance of some economic interaction, say a potential exchange, one individual is relatively more informed. The focus is on how the outcomes in such an incomplete information environment may differ from that wherein information is complete. The regret game has, in a sense, exactly the opposite core feature of `ex-post asymmetric information'. At the onset everyone possesses the same information; but after actions have been taken individuals may possess differing levels of information. A particular type of market failure can emerge:\ all individuals take a decision that is collectively suboptimal so as to avoid being the one with more, not less, information ex-post.

We then move to testing our theory experimentally. In a big-picture sense, we begin by eliciting certainty equivalents when ex-post information is total and compare it with the elicited certainty equivalents when ex-post information depends on choice. We find strong support for Theorem \ref{thm:lessInfo} in that many subjects make different choices when the only difference is the ex-post information supplied. (Theorem \ref{thm:regretEquivalent} ensures that our mechanism is incentive compatible.)  The results from these initial decisions also allow us to classify participants into those who are regret averse and those who are not,\footnote{As further discussed in Section \ref{experiment1}, the idea that the expectation of feedback affects anticipated regret and thus behaviour has been studied in the psychology literature \cite{Zeelenberg:1996}. Yet ours is the first paper that identifies regret averse individuals by manipulating feedback and using an incentive-compatible mechanism.} and to calibrate the parameters used in the second part of the experiment wherein our participants play a version of the regret game.

More specifically, we run two experiments both using a within-subject design. The first experiment consists of two parts. In the first part, we elicit participants' preferences over a sure amount of money and a risky lottery under two different information conditions -- one where participants learn the risky lottery's outcome even if they do not choose it, and one where they do not learn the risky lottery's outcome unless they choose it.
In the second part of the experiment, participants are matched in pairs and play the regret game. They must choose between a sure amount of money and a lottery. If they do not choose the lottery, they will learn its outcome only if their partner chose the lottery.

We find that, when the regret game has a unique equilibrium in dominant strategies, the vast majority of participants choose it whether they are regret averse or not. When the regret game is a game of coordination, regret averse participants try to coordinate with their partner. This supports our model's predictions. We observe a positive and highly significantly impact of beliefs on choice for regret averse participants. When we focus on the last iteration of the game, the effect is even stronger. We also observe a positive impact of beliefs on choice for non regret averse participants, but the effect is smaller and only marginally significant.

These results indicate that regret aversion can drive coordination. However, in our setting there may have been two additional drivers of coordination:\ preferences for conformism \cite{CharnessRigottiRustichini:2017:GEB,CharnessNaefSontuoso:2019:JET} and inequity averse preferences \cite{FehrSchmidt:1999:QJE}. To eliminate these potential confounds we ran a second experiment. The second experiment consists of two parts. In the first part, participants choose between a sure amount of money and a risky lottery, with subjects revealing to us their certainty equivalent for the risky lottery. Subjects are later asked whether they want to find out how much they would have earned had they chosen the risky lottery. If they choose not to find out, they forgo a small amount of money. This procedure allows us to classify participants into those who are regret averse and those who are not. In the second part of the experiment, participants are matched in pairs and play a variant of the regret game:\ they must choose whether they want to find out the risky lottery's outcome or not, but they can avoid finding out only if their partner also decides not to find out.

Again we find that regret averse participants try to coordinate with their partner. The probability that they choose what they think their partner chose is significantly higher than the probability that they choose the alternative. All regret averse participants who believe that their partner chose to avoid information, choose to avoid information. In contrast, non regret averse participants play the dominant strategy, and some of them do so under the belief that their partner chose the alternative option. Thus, once the aforementioned confounds are removed by design, we still find strong support for our model's predictions.

Regret aversion dates back to the classic works of \citet{Bell:1982} and \citet{LoomesSugden:1982:EJ}. However, these and subsequent studies of regret aversion assume that ex-post information is total.
In order for any behavioural bias to ``have bite'' after a decision has been made, information about foregone alternatives is required. We show how to model behavioural agents with biases that ``kick in'' after decisions have been taken, and further to document how choice can vary with uncertainty over what environment will be faced ex-post (in addition to the standard uncertainty associated with the outcomes of each choice). While we have focused on regret, we are hopeful the machinery will be useful to address other biases too.

While a number of papers explore how regret impacts choice in individual decision problems, the impact of regret averse preferences in strategic environments is extremely limited. A notable exception is \citet{Filiz-OzbayOzbay:2007:AER} who incorporate anticipated regret into the preferences of agents partaking in a sealed-bid auction.\footnote{There is a large literature that studies the connection between dynamics based on ``regret minimisation'' and convergence to Nash equilibria and correlated equilibria \cite{Aumann:1974:JME}. See for example, \citet{FosterVohra:1997:GEB}, \citet{HartMas-Colell:2000:E}, and the textbook \citet{Cesa-BianchiLugosi:2006:}.} However, unlike in our paper, individuals in their environment either \emph{always} find out the outcome of the foregone alternative or never find out.  That is, the informational environment is set exogenously and does not depend endogenously on the behaviour of individuals. In our set up, anticipated regret can serve as a coordination device.

In models of \emph{social learning} \cite{BikhchandaniHirshleifer:1992:JPE,Banerjee:1992:QJE}, the behaviour of others generates information but is payoff irrelevant. Mapping this to the Ann and Barry story, individuals choose sequentially basing their choice on a weighted combination of prior belief and the choices of those who went before. In a model of \emph{social interaction} (a game), someone else's choice is payoff relevant, but does not generate information.  The regret game is a behavioural game that, in a sense, lies somewhere between the two settings above:\ someone else's choice is payoff relevant, but only because it can affect ex-post information and this can be potentially harmful to a behavioural individual.

The two papers closest to ours are \citet{Benabou:2013:RES} and \citet{CooperRege:2011:GEB}. Using preferences for late-resolution of uncertainty from \citet{KrepsPorteus:1978:E}, \citet{Benabou:2013:RES} presents a dynamic model that addresses the harmful issue of ``groupthink''. In the model each individual's payoff depends on the effort level of everyone (including his own) and the realisation of a random variable.  The key feature is the inclusion of \emph{anticipatory utility} experienced from thinking about one's future prospects. The more positive the forecast, the better for the individual. This allows for multiple equilibria including, for example, one in which everyone in the population collectively ignores a negative public signal about the random variable. Such delusions can persist because individual $j$'s informational decision and effort choice can affect the risks of individual $i$'s psychological payoff, leading $i$ to make a different informational decision than he otherwise would. In our model by contrast, $j$'s choice of lottery can impact the informational environment that $i$ faces, changing $i$'s psychological payoffs and leading $i$ to make a different risk-taking decision than he would in isolation.

\citet{CooperRege:2011:GEB} present a model of ``social regret'' wherein the regret from a choice that turned out suboptimal is dampened if others chose the same. An individual tells himself that his decision could not have been too wrong if many others acted the same. Misery loves company. This model's key feature is the belief that an individual assigns to others choosing an alternative. The more likely another individual is to choose an alternative, the greater the expected regret from not choosing that alternative. Results from laboratory experiments confirm their hypothesis that social regret is a powerful force. In our model, an individual's choice imposes an externality on others by changing the information environment that they may face. In \citet{CooperRege:2011:GEB}, the effect of social regret is magnified by the number of others who make a different choice, but social regret is less intense when others make the same choice.\footnote{A related literature is that on envy or ``keeping up with the Joneses'', according to which people care not only about the absolute value of their own consumption, but also about the average (or per capita) level of consumption in the economy \cite{Gali:1994:JMCB}.}

The remainder of the paper is structured as follows. Section \ref{MODEL} presents the model.
Section \ref{experiments} describes the designs and results of our two experiments. Section \ref{CONCLUSION} concludes.

\section{The Model}\label{MODEL}

Ex-post information feedback is an integral part of the regret story. If the outcome of a foregone alternative is never learned, then how can it be regretted? In subsection \ref{subsec:RegPref}, we build on this insight to formally explore the mechanics of regret aversion. In effect, a regret averse individual `zeroes-in' on the best-performing lottery in each state, creating a state-dependent reference point that can only be matched but never exceeded. Formally, identifying each state of nature by its best-performing lottery generates a \emph{regret-relevant information partition} of the state space. However, implicit in this construction is that alternatives, in particular the best-performing lottery, in each state will be learned about even if unchosen. We consider how preference reversals may occur when the regret-relevant partition that is faced depends on choice. {Using our example from Section \ref{INTRODUCTION}, we imagine that Ann only learns the outcome of the risotto if she orders it  (since the spaghetti dish is a sure thing, it is always known)}.\footnote{We emphasise that risk free lotteries therefore play a very important role for regret averse individuals.} This asymmetry in ex-post information, and hence regret, increases the relative benefit of choosing spaghetti as it is a safe haven from regret.

In Subsection \ref{subsec:RegPrefUncertainInfo}, we extend the environment to one in which the information that will be learned ex-post is unknown in advance. That is, in addition to the `standard' uncertainty captured by risky lotteries, there is an additional layer of uncertainty due to not knowing what regret-relevant information partition will be faced ex-post. Regret may or may not be experienced. {Again using our example, we suppose that there is a possibility that Ann will learn of the risotto's quality even if she orders the spaghetti (perhaps a she will observe someone at a different table order it). Ordering the spaghetti no longer provides full insurance from regret.} In Subsection \ref{subsec:strategicSetting} we extend to an environment where there are many regret averse individuals, and the regret-relevant information partition that is faced by an individual depends probabilistically upon her own choice \emph{and} the choices of others. {We suppose that Ann has a friend, Barry, who joins her for dinner. If Barry orders the risotto, then Barry's choice means that Ann will learn the risotto's quality.} In this case, Barry's behaviour imposes a negative externality on Ann's ex-post psychological payoff. This is a game in which Barry does not alter Ann's payoff directly, but rather he impacts the ex-post informational environment that Ann will face.

\subsection{The Decision Environment}\label{subsec:RegPref}

Let $\Omega$ denote a finite state space with typical element given by $\omega$. Uncertainty is captured by a probability measure $\mathbb{P}$ defined on $2^{\Omega}$, where $\mathbb{P}[\set{\omega}] >0$ for all $\omega \in \Omega$.\footnote{Since $\Omega$ is finite there are no technical headaches.} There is a choice set, $\mathcal{L}$, containing $n$ risky lotteries, labelled $\ell_{1}, \ell_{2}, \dots, \ell_{n}$, and a safe (risk-free) lottery, $\ell_S$. The outcome of lottery $\ell$ in state $\omega \in \Omega$ is denoted by $\ell(\omega)$. For simplicity we assume that each risky lottery, $\ell_{i}$, $i = 1, \dots, n$, is an independent Bernoulli random variable with outcome $\underline{\ell}_{i}$ occurring with probability $1-p_{i} \in (0, 1)$ and outcome $\overline{\ell}_{i}$ occurring with probability $p_{i}$. We further assume that there are no payoff ties and that outcomes are structured such that $\max_{i}{\underline{\ell}_{i}} < \ell_{S} < \min_{i}{\overline{\ell}_{i}}$.
Therefore there are exactly $2^{n}$ states in $\Omega$. Note that the risk free lottery, $\ell_S$, is the lottery with highest return in only one state; in all other states, at least one of the risky lotteries will outperform it.\footnote{\label{fn:risk-free}Since the $n$ risky lotteries have uncorrelated returns, then the probability that the risk-free lottery is the best-performing lottery equals $\prod_{i=1}^{n}(1-p_{i})$. This tends to zero as $n$ gets large.}

Let $u(\cdot)$ be a real-valued \emph{choiceless utility} function defined on $\mathcal{L} \times \Omega$ that satisfies the usual conditions.\footnote{The term ``choiceless utility'' was introduced by \citet{LoomesSugden:1982:EJ}. It is so-called as it is the utility experienced if the decision maker is simply \emph{assigned} lottery $\ell$ and the resulting state is $\omega$} Let $R(\ell; \omega)$ capture the experienced regret in state $\omega$ when lottery $\ell$ was chosen; formally it is a function of the difference between `choiceless utility from what turned out to be the best possible decision' and `choiceless utility from the decision made'.\footnote{This is different to the set up of \citet{LoomesSugden:1982:EJ}, who view regret as stemming from pairwise comparisons. Our formulation is based on \citet{Sarver:2008:E}, where the comparison lottery is that which performed best in the realised state.}
The total utility experienced by a decision maker in state $\omega \in \Omega$, when lottery $\ell \in \mathcal{L}$ is chosen, is denoted by $u^{T}$ and is defined as
\begin{align}
u^{T}\big(\ell(\omega)\big) &=  u\big(\ell(\omega)\big) - R(\ell; \omega) \label{eq:regret1}\\
&=  u\big(\ell(\omega)\big) - \kappa \big( \max_{\ell' \in \mathcal{L}}u(\ell'(\omega)) - u(\ell(\omega)) \big) \label{eq:regret2} \nonumber
\end{align}
where parameter $\kappa$ $(\geq0)$ is the coefficient of regret aversion that is assumed to be state-independent.

A decision maker compares the expected total utility of all the lotteries. Letting $\mathbb{E}_{\omega}$ denote the expectation operator with respect to $\mathbb{P}$, we can state the decision maker's optimisation problem as,
\begin{equation}
\max_{\ell \in \mathcal{L}} \mathbb{E}_{\omega}\Big[u^{T} \big(\ell(\omega)\big)\Big] \label{eq:Dproblem}
\end{equation}

We now state our first result.

\begin{theorem}\label{thm:regretEquivalent}
Suppose the outcome of all lotteries will always be learned ex-post.  Then, for every pair of lotteries $\ell_{i}$ and $\ell_{j}$ in $\mathcal{L}$,
\[
\mathbb{E}_{\omega}\Big[\big(u (\ell_{i}(\omega)\big)\Big] \geq \mathbb{E}_{\omega}\Big[\big(u (\ell_{j}(\omega)\big)\Big] \iff \mathbb{E}_{\omega}\Big[\big(u^{T} (\ell_{i}(\omega)\big)\Big] \geq \mathbb{E}_{\omega}\Big[\big(u^{T}(\ell_{j}(\omega)\big)\Big]
\]
\end{theorem}

The proof is in the Appendix. Theorem \ref{thm:regretEquivalent} states that two decision makers, one with standard preferences ($\kappa=0$) and the other regret averse ($\kappa>0$), are indistinguishable in their choice behaviour. While the proof is almost immediate, it is important for our experimental purposes as it ensures that the incentive compatible mechanism that we use in our experiments, the BDM \citep{BDM}, remains incentive compatible under regret averse preferences. We note that while Theorem \ref{thm:regretEquivalent} is stated for Bernoulli lotteries, it is easily extended to an environment with more general lotteries provided everything remains finite.

While interesting, Theorem \ref{thm:regretEquivalent} does require that the outcome of every lottery is learned ex-post. However, while a lottery's outcome will always be learned when it is chosen, the same may not be true for unchosen lotteries.  In such environments, what really matters to a regret averse individual is not the best performing lottery in each state, but the best performing lottery in each state that is learned about. The standard regret framework is insufficient for these purposes and needs to be extended to allow for the possibility that the information available ex-post depends on choice. We develop such an extension now.

Begin by relabelling the lotteries in such a way that $\overline{\ell}_{1} > \overline{\ell}_{2} > \overline{\ell}_{3} > \dots > \overline{\ell}_{n-1} > \overline{\ell}_{n} > \ell_{S}$. Now, labelling lottery $S$ by $n+1$, for each $i =1, \dots, n+1$, define the event $F(i)$ as follows:
\begin{equation}
F(i) := \set{\omega \in \Omega \,\, \Big| \,\, \ell_{i}(\omega) =  \max_{\ell_j\in\mathcal{L}} \ell_{j}(\omega)} \label{eq:fullPartition}
\end{equation}
In words, event $F(i)$ is the set of states upon which lottery $\ell_{i}$ is the best-performing. {From our assumptions on the structure of lottery outcomes each $F(i)$ is nonempty and the collection $\set{F(j)}_{j = 1}^{n+1}$ forms a partition of the state space $\Omega$.} In words, a regret-averse individual creates a reference point for every state, where the state-dependent reference point is the outcome of the best-performing lottery. Such a reference point can only be matched but never exceeded.

The partition given in \eqref{eq:fullPartition} assumes that ex-post information will be total. But what happens when the ex-post information the decision maker receives depends on choice? That is, consider the event $F(j)$ but suppose that the environment is changed such that the outcome of $\ell_{j}$ in state $\omega \in F(j)$ is not learned when lottery $\ell_{k}$ is chosen. Clearly the state dependent reference point $\ell_{j}(\omega)$ is no longer valid. We assume that it is replaced with the payoff to the best-performing lottery that is learned about in state $\omega$.

When lottery $\ell_{k}$ is chosen, we let $\mathcal{O}_{k}\subseteq \mathcal{L}$ denote the set of lotteries whose outcomes are observed.\footnote{Note that this assumes that, conditional on choosing lottery $k$, the set of lotteries whose outcomes are learned does not vary with the realised state. While this generalisation is easily incorporated, the simpler version is sufficient for our purposes.}  Clearly $\mathcal{O}_{k}$ is non empty for all $k$ as the outcome of the chosen lottery is always learned. To incorporate ex-post information that depends on choice, we amend \eqref{eq:fullPartition} above, and define $F_k(j)$ as follows.

\begin{equation}
F_k(j) := \set{\omega \in \Omega \,\, \Big| \,\, \ell_{j}(\omega) =  \max_{\ell_h\in\mathcal{O}_{k}} \ell_{h}(\omega)} \label{eq:partition}
\end{equation}
That is, $F_k(j)$ is the set of states where lottery $j$ is the best performing lottery that is learned about conditional on lottery $k$ being chosen. Unlike the events defined in \eqref{eq:fullPartition}, it is possible for $F_k(j)$ to be empty. This is despite the fact that every lottery is the best-performing in at least one state.

So, for every lottery $k$ the decision maker associates a partition of $\Omega$ given by $\pi_{k} = \set{F_{k}(j)}_{\ell_{j}\in \mathcal{L}}$. We call this the \emph{regret relevant information partition} associated with lottery $k$.  Note that when the outcomes of all lotteries can be learned ex-post, $F_{k}(j)=F (j)$, for every lottery $k$.\footnote{In this case the collection $\set{F(i)}_{i=1}^{n+1}$ forms the \emph{fundamental regret relevant information partition} that we denote by $\pi_{0}$.}

We define an \emph{ex-post information environment}, $\Pi$, as the collection of regret relevant information partitions, one for each lottery. That is, $\Pi = \set{\pi_{k}}_{k=1}^{n+1}$. We now show how to define a partial order on the collection of all ex-post information environments, that allows us to rank them in terms of greater ex-post `informativeness'.

\begin{defn}\label{def:moreInfo}
We say that ex-post information environment $\Pi' = \set{\pi_{k}'}_{k=1}^{n+1}$ is \emph{more informative} than ex-post information environment $\Pi'' = \set{\pi_{k}''}_{k=1}^{n+1}$ if for every $k = 1, \dots, n+1$, regret relevant information partition $\pi_{k}'$ is as fine as regret relevant information partition $\pi_{k}''$, and $\pi_{j}'$ is strictly finer than $\pi_{j}''$ for at least one lottery $j$.\footnote{Given a set $S$ and two partitions of $S$, $\rho_{1}$ and $\rho_{2}$, we say that $\rho_{1}$ is finer than $\rho_{2}$ if every element $\alpha$ of $\rho_{1}$ is a subset of some element of $\rho_{2}$.}
\end{defn}

We now show that a more informative ex-post information environment is not desirable for a regret averse individual.

\begin{theorem}\label{thm:lessInfo}
Consider two ex-post information environments $\Pi'$ and $\Pi''$ such that $\Pi'$ is more informative than $\Pi''$. Then a regret averse individual prefers ex-post information environments $\Pi''$ to $\Pi'$, in the sense that the expected total utility associated with a given choice of lottery is never higher in $\Pi'$.
\end{theorem}

Theorem \ref{thm:lessInfo} is very intuitive. A regret averse individual constructs a reference point for every (lottery,state) pair, that is given by the best-performing lottery that is learned about. A higher reference point is bad as the difference between the reference point and the chosen lottery's outcome can only increase, thereby increasing the regret that is experienced. By the definition of more informativeness, the reference point is never decreasing and strictly increases for at least one (lottery,state) pair. Since the choiceless utility experienced does not depend on the ex-post information environment and a more informative environment brings with it more regret, then such an environment is not attractive to a regret averse individual.

We now show how to encode the example of Ann deciding between restaurants from Section \ref{INTRODUCTION} into our framework.

\begin{ex}
Regret averse Ann must decide between two pizza-serving restaurants, $R$ and $S$. There are two states of the world, $\omega_{1}$ and $\omega_{2}$, with the probability of state $\omega_{1}$ given by $p \in (0, 1)$. Restaurant $R$ is risky and its pizza brings payoff ${\underline{\ell}_{R}}$ in state $\omega_{1}$ and payoff ${\overline{\ell}_{R}}$ in state $\omega_{2}$. Restaurant $S$ is the safe choice, with its pizza bringing payoff $\ell_{S}$ in both states. Payoffs are structured such that ${\underline{\ell}_{R}}  < \ell_{S} < {\overline{\ell}_{R}}$.

There are two information environments $\Pi'$ and $\Pi''$. Under $\Pi'$, the outcome of both lotteries is always learned ex post, while under $\Pi''$, the outcome of lottery $R$ is learned only if chosen. Therefore, we have
\begin{align*}
\text{Under } \Pi': & \hspace{.2in} \mathcal{O}_{\ell_{R}} = \set{\ell_{R}, \ell_{S}}, \mathcal{O}_{\ell_{S}} = \set{\ell_{R}, \ell_{S}} \text{ and }  \pi'_{\ell_{R}}  = \set{\set{\omega_{1}}, \set{\omega_{2}} }= \pi'_{\ell_{S}} \\
\text{Under } \Pi'': & \hspace{.2in} \mathcal{O}_{\ell_{R}} = \set{\ell_{R}, \ell_{S}}, \mathcal{O}_{\ell_{S}} = \set{\ell_{S}} \text{ and } \pi''_{\ell_{R}}  = \set{\set{\omega_{1}}, \set{\omega_{2}}} \neq \pi''_{\ell_{S}}  = \set{\set{\omega_{1}, \omega_{2}}}
\end{align*}
Let us now consider the expected total utility associated with each choice of lottery. That is, we evaluate $u^{T}(\ell, \Pi)$, where we note the explicit dependence on the information environment $\Pi$. We have
\begin{align*}
u^{T}(\ell_{R}, \Pi') &= p\Big(u({\underline{\ell}_{R}}) - \kappa\big(u({\underline{\ell}_{R}}) - u(\ell_{S})\big)\Big) &+ (1-p)\big(u({\overline{\ell}_{R}})\big)\\
u^{T}(\ell_{R}, \Pi') &= pu({\ell}_{S}) &+ (1-p)\Big(u(\ell_{S}) - \kappa\big(u(\ell_{S}) - u({\overline{\ell}_{R}})\big)\Big)
\end{align*}
while
\begin{align*}
u^{T}(\ell_{R}, \Pi'') &= p\Big(u({\underline{\ell}_{R}}) - \kappa\big(u({\underline{\ell}_{R}}) - u(\ell_{S})\big)\Big) &+ (1-p)\big(u({\overline{\ell}_{R}})\big)\\
u^{T}(\ell_{R}, \Pi'') &= pu({\ell}_{S}) &+ (1-p)u(\ell_{S})
\end{align*}
where we note that under information environment $\Pi''$, by choosing $\ell_{S}$ Ann is completely insured against regret. Note that $S$ has become relatively more attractive to Ann in $\Pi''$.
\end{ex}

We conclude this section by reiterating the main observation. Models of regret aversion assume that the decision maker identifies each state by its best-performing lottery. But this is not always possible. When this is the case the regret averse decision maker identifies each state by the best-performing lottery that is learned about. Finally, what lottery outcomes are learned and what are not may be choice dependent. It then immediately follows that varying the information environment can impact (optimal) choice. In the next subsection we will allow for the possibility that multiple regret-relevant information partitions can be associated with each lottery choice. That is, there will be further uncertainty about what ex-post information environment will be faced.

\subsection{Uncertainty Over the Ex-post Information Structure}\label{subsec:RegPrefUncertainInfo}

In this section we allow for the possibility that the outcomes of unchosen lotteries may be learned about or may not be learned about. That is, there may be more than one regret relevant partition associated with a lottery choice. To illustrate the effect this can have on optimal choice we limit attention to the environment where the choice set $\mathcal{L}$ only contains one risky lottery $\ell_{r}$ and the risk free lottery $\ell_{S}$. With only one risky lottery the state space is $\set{\omega_{1}, \omega_{2}}$, where $\omega_{1}$ and payoff $\bar{\ell}_{r}$ occur with probability $p \in (0, 1)$, and $\omega_{2}$ and payoff $\underline{\ell}_{r}$ occur with probability $1-p$.

The risk-free lottery has the property that its outcome will always be known whether it was chosen or not. The same need not be true of the risky lottery. We let $q \in [0, 1]$ be the probability that an agent learns the outcome of the risky lottery, $\ell_{r}$, conditional on choosing the safe lottery, $\ell_{S}$. That is, conditional on choosing the risk-free lottery, with probability $q$ the individual faces regret relevant partition $\pi_{0} = \set{F(S), F(\ell_{r})}$, and with probability $1-q$ he faces partition $\pi_{S}' = \set{F'(S)}$ where $F'(S) = \Omega$. Utility is then given by
\begin{equation}\label{eq:URstate0}
u^{T}(\ell_{S}, \omega, q) = \left\{ \begin{array}{rl}
 u(\ell_{S}) - q\kappa\big(u(\overline{\ell}_{r}) - u(\ell_{S}) \big), &\mbox{ if } \omega = \omega_{1} \\
 u(\ell_{S}), &\mbox{ if } \omega = \omega_{2}
       \end{array} \right.
\end{equation}
and
\begin{equation}\label{eq:URstate1}
u^{T}(\ell_{r}, \omega, q) = \left\{ \begin{array}{rl}
 u(\overline{\ell}_{r}) , &\mbox{ if } \omega = \omega_{1} \\
 u(\underline{\ell}_{r}) - \kappa\big( u(\ell_{S}) - u(\underline{\ell}_{r}) \big), &\mbox{ if } \omega = \omega_{2}
       \end{array} \right.
\end{equation}

Both the risky lottery, $\ell_{r}$, and the risk-free lottery, $\ell_{S}$, bring with them a benefit and a cost. The benefit is the direct choiceless utility associated with each; the cost is the psychological penalty that may be incurred in the event your choice is not optimal in the realised state. For the risky lottery, $\ell_{r}$, both the expected benefit and expected cost are fixed. For the risk-free lottery, $\ell_{S}$, the expected benefit is fixed but the expected cost is not. It may or may not be incurred.

Using \eqref{eq:URstate0} and \eqref{eq:URstate1}, it is simple to calculate when the risky option is preferable to a regret averse individual. To make things as clear as possible, we normalise the choiceless utility function $u$ so that $u(\underline{\ell_{r}}) = 0$ and $u(\ell_{S}) = 1$. With this, the condition for the risky lottery $\ell_{r}$ to be preferred to the risk-free lottery $\ell_{S}$ reduces to
\begin{equation}\label{eq:EURcondition}
u(\overline{\ell}_{r}) \geq \frac{1}{p}\left(\frac{1+\kappa \big(1-p(1-q)\big)}{1+q\kappa}\right)
\end{equation}

Expression \eqref{eq:EURcondition} is bookended by two important cases. When $q=1$ the decision maker will learn the outcome of the risky option no matter what.  Here, there is no distortion to the threshold rule relative to standard preferences, in that,
\begin{equation}\label{eq:EURconditionq=1}
u(\overline{\ell_{r}}) \geq \frac{1}{p}
\end{equation}

When $q = 0$ however, the decision maker knows that he will definitely \emph{not} learn the outcome of the risky option unless he opts for it. Then \eqref{eq:EURcondition} becomes,
\begin{equation}\label{eq:EURconditionq=0}
u(\overline{\ell}_{r}) \geq \frac{1}{p}\big( 1+\kappa(1-p) \big)
\end{equation}

Given that $\kappa>0$ and $p \in (0, 1)$, the inequality in \eqref{eq:EURconditionq=0} is a more demanding condition on the risky lottery $\ell_{r}$ than that in \eqref{eq:EURconditionq=1}. That is, when a regret averse decision maker will certainly not find out the realisation of the risky option without choosing it, he requires it to have a more desirable payoff distribution. (It can be checked that the RHS of expression in \eqref{eq:EURcondition} is strictly decreasing in $q$ over the interval $[0, 1]$.)

The reason for the discrepancy above is that the risk-free lottery can provide insurance against regret. When $q=0$, the risk-free lottery provides complete insurance against regret. There is complete asymmetry in anticipated regret:\ if the decision maker chooses the risky option, he knows he will be able to make an ex-post comparison and feel regret if the risky option is not successful, whereas if he chooses the risk-free lottery, he knows he will not be able to make such a comparison. Because of the insurance against regret offered by the risk-free lottery, the outcome of the risky lottery in the event of a success must be high enough to tempt the decision maker away from the security of the risk-free lottery. Ignorance is bliss. On the other hand, when $q=1$, the risk-free lottery offers no insurance against regret. The decision maker knows he will learn the outcome of the risky lottery whether he chooses it or not. The regret considerations are symmetric and cancel each other out, and the individual's condition is the same as if he were regret neutral.

Intuitively, the benefit that the risky lottery must yield in order to be chosen becomes lower as $q$ increases due to the reduction in anticipated regret.  As $q$ decreases (i.e., the likelihood of making an {ex-post} comparison in the case the risky option is not chosen reduces) the agent is increasingly - from an {ex-ante} perspective - insured against regret.  And because of this insurance against potential regret, the risky lottery's outcome must increase to tempt the agent away from the safe option.

In the next subsection, we extend the setting to one with multiple regret averse agents, and we suppose that $q$ is endogenously determined by the choices of others. In particular, we assume that $q$ is increasing in the number of other agents who choose the risky lottery. This seemingly minor amendment turns a series of single-person decision problems into a multi-player behavioural game.

\subsection{Strategic Setting: The Regret Game}\label{subsec:strategicSetting}

\subsubsection{The set up}

We stick with the 2-state 2-lottery environment from the previous subsection. Suppose now that there are $N$ decision makers, each of whom is regret averse, i.e., with preferences represented by $u^{T}$ with $\kappa>0$. The key modelling feature we introduce is that an individual's likelihood of learning about the risky outcome conditional on choosing the risk-free lottery is a function of the behaviour of others.

To capture the above, we define a symmetric $N$-player (simultaneous-move) behavioural game with common action set $A$ and common utility function $u^{T}$.\footnote{This is an abuse of notation as the domain on which utility is defined will now be extended to include the behaviour of others.} We identify the action set $A$ with the 2-element choice set $\mathcal{L} = \set{\ell_{r}, \ell_{S}}$. Each player $i$, chooses an action $a_{i} \in A = \set{\ell_{r}, \ell_{S}}$, and has utility function $\Ui^{T} : \mathbf{A} \times \set{\omega_{1}, \omega_{2}} \to \mathbb{R}$, where $\mathbf{A} := \prod_{j =1}^{n} A_{j}$, with typical element $\mathbf{a} = (a_{1}, \dots, a_{n})$.  From player $i$'s perspective, a pure action profile $\mathbf{a} \in \mathbf{A}$ can be viewed as $(a_{i}, \mathbf{a}_{-i})$, so that $(\hat{a}_{i}, \mathbf{a}_{-i})$ will refer to the profile $(a_{1}, \dots, a_{i-1}, \hat{a}_{i}, a_{i+1}, \dots, a_{n})$, i.e., the action profile $\mathbf{a}$ with $\hat{a}_{i}$ replacing $a_{i}$. The utility function of agent $i$ is as defined in \eqref{eq:URstate0} and \eqref{eq:URstate1} save one difference:\ the probability that $i$ learns the outcome of the risky lottery when it is not chosen, denoted $q_{i}$, depends on the behaviour of the other agents. Formally,
\begin{equation}\label{eq:UiRstate0}
\Ui^{T}\big((\ell_{S}, \mathbf{a}_{-i}), \omega\big) = \left\{ \begin{array}{rl}
 u(\ell_{S}) - q_{i}(\mathbf{a}_{-i})\kappa\big(u(\overline{\ell}_{r}) - u(\ell_{S}) \big), &\mbox{ if } \omega = \omega_{1} \\
 u(\ell_{S}), &\mbox{ if } \omega = \omega_{2}
       \end{array} \right.
       \end{equation}
and
\begin{equation}\label{eq:UiRstate1}
\Ui^{T}\big((\ell_{r}, \mathbf{a}_{-i}), \omega\big) = \left\{ \begin{array}{rl}
 u(\overline{\ell}_{r}) , &\mbox{ if } \omega = \omega_{1} \\
 u(\underline{\ell}_{r}) - \kappa\big( u(\ell_{S}) - u(\underline{\ell}_{r}) \big), &\mbox{ if } \omega = \omega_{2}
       \end{array} \right.
\end{equation}
where $q_{i}(\mathbf{a}_{-i})$
is assumed to be strictly increasing in the number of others that choose $\ell_{r}$.\footnote{Assuming that $q_{i}$ is increasing seems natural to us. The more individuals who choose the risky lottery, the harder it ought be not to learn about its performance ex-post. But while it seems implausible that the function $q_{i}$ would be decreasing at any point, we leave its precise functional form unspecified. One can imagine settings in which $q_{i}$ is linear, concave, and convex. One can even envisage settings where $q_{i}$ is step function, in that the outcome of $\ell_{r}$ cannot be avoided once \emph{enough} individuals have chosen it.} That is, abusing notation somewhat, we let $\abs{\mathbf{a}} = \abs{(a_{i}, \mathbf{a}_{-i})} := \#\set{j \neq i : a_{j} = \ell_{r} }$. Then for any two profiles $\mathbf{a}'$ and $\mathbf{a}''$ we have $q_{i}(\abs{\mathbf{a}'}) > q_{i}(\abs{\mathbf{a}''})$ if and only if $\abs{\mathbf{a}_{-i}'} > \abs{\mathbf{a}_{-i}''}$. Finally, we assume that $q(0) = 0$ and $q(N-1) = 1$.

We emphasise that, while player $i$'s utility depends on the choices of everyone, and hence the set up is a strategic game, the dependence is not direct. Rather it manifests through the likelihood that player $j$'s choice of lottery will impact player $i$'s ex-post regret relevant partition. It is individual $j$'s risk taking that can generate information for individual $i$, in turn altering $i$'s psychological payoffs, in turn altering the relative benefits of each choice. We now characterise the set of equilibria to this set up:\ showing how individual psychological motives can lead to socially interdependent decisions.

\subsubsection{Equilibria}

The above defines the Regret Game. There are three classes of the game, depending upon the parameters. When
$u(\overline{\ell}_{r})<1/p$, it is a dominant choice for each player to choose the safe lottery $\ell_{S}$. Similarly, when $u(\overline{\ell}_{r}) > \frac{1}{p}\left( 1+\kappa(1-p) \right)$, choosing the risky lottery $\ell_{r}$ is the dominant action. For intermediate values of $u(\overline{\ell}_{r})$ however, the game is one of coordination and has two pure-strategy Nash equilibria:\ all players choose $\ell_{S}$, and all players choose $\ell_{r}$.\footnote{There is also a completely mixed strategy equilibrium but we ignore it as it is very unstable.} Theorem \ref{thm:Nashmany} below states this formally.

\begin{theorem}\label{thm:Nashmany}
In the regret game,
\begin{enumerate}
\item\label{condition:Nashmany1}
When $u(\overline{\ell}_{r}) < \frac{1}{p}$, uniform adoption of the risk-free lottery, $\ell_{S}$, is the unique (dominant) pure strategy Nash equilibrium.

\item\label{condition:Nashmany2}
When $u(\overline{\ell}_{r}) > \frac{1}{p}\left( 1+\kappa(1-p) \right)$, uniform adoption of the risky lottery, $\ell_{r}$, is the unique (dominant) pure strategy Nash equilibrium.

\item\label{condition:Nashmany3}
When $u(\overline{\ell}_{r}) \in \big[\frac{1}{p}, \frac{1}{p}\left( 1+\kappa(1-p) \right) \big]$, uniform adoption of the risk-free lottery, $\ell_{S}$, is a pure strategy Nash equilibrium and uniform adoption of the risky lottery, $\ell_{r}$, is a pure strategy Nash equilibrium

\end{enumerate}
\end{theorem}

Theorem \ref{thm:Nashmany} can be understood as follows. While the risky lottery, $\ell_{r}$, brings a guaranteed utility, for every (other) individual who chooses the risky lottery, the expected utility associated with the risk-free lottery is decreasing. The reason for this being that the likelihood of learning about the alternative, and hence experiencing regret, is going up. Thus, for the parameters of case \ref{condition:Nashmany3}, we have a slightly unusual coordination game in that the number of others who choose $\ell_{r}$ is decreasing the associated net utility of choosing $\ell_{S}$ without improving the net utility of choosing $\ell_{r}$.

\subsubsection{Welfare and equilibrium selection}

It is interesting to compare the (common) expected utility levels at each equilibrium in case \ref{condition:Nashmany3}. From  \eqref{eq:UiRstate0} and  \eqref{eq:UiRstate1} we can compute that uniform adoption of the risky lottery, $\ell_{r}$, is Pareto optimal if and only if $u(\overline{\ell}_{r}) \geq \frac{1}{p}\big( 1+\kappa(1-p) \big)$. But this is precisely the threshold at which individuals would always choose the risky lottery in any case. Thus, for parameters such that the regret game is a coordination game (case \ref{condition:Nashmany3}), coordinating on the risk-free lottery is Pareto-optimal (and always preferred ex-ante).

There is a large literature addressing the tension that exists between the multiple equilibria in coordination games. The most commonly studied environment is a large-population binary-action game (e.g., the Stag Hunt) where the Pareto efficient equilibrium does not coincide with the \emph{safe} risk-dominant equilibrium.\footnote{Probably the best known experimental test of the efficiency vs safety issue is \cite{Van-HuyckBattalio:1990:AER} who study human behaviour in the \emph{minimum effort game}, of which the Stag Hunt is a simple version.} Existing equilibrium selection techniques - be they evolutionary like \emph{stochastic stability} \citep{KandoriMailath:1993:E, Young:1993:E} or higher-order belief-based like \emph{global games} \citep{CarlssonDamme:1993:E, MorrisShin:2003:} - favour the equilibrium that is more difficult to destabilise (i.e., the risk-dominant one). For our regret game, simple algebra shows that $q \geq q^{\star} = 1/2$ is the threshold at which the risky lottery becomes most desirable. But if we imagine a \emph{steep} specification of $q$ (in that $q_{i}(\abs{\mathbf{a}}) \approx 1$ when $\abs{\mathbf{a}} \geq 1$), then uniform adoption of the risky lottery would be the prediction. But that means the regret game has the quite curious property that the risk-free Pareto dominant outcome need not be risk-dominant \citep{HarsanyiSelten:1988:}.

\subsubsection{Extensions}

In what follows, we briefly describe some extensions of our model. Further details can be found in Appendix \ref{extensions}.

In the regret game above we have assumed that everyone is regret averse. However, in reality it is possible that only some proportion of a population are regret averse ($\kappa > 0$) and the remainder are regret neutral ($\kappa = 0$). The first extension in Appendix \ref{hetero-prefs} considers the case in which some proportion of the population are similarly regret averse and the remaining proportion are regret neutral. The preferences over lotteries for regret neutral individuals
are fixed no matter the information feedback. So these individuals always choose the lottery with the highest expected (choiceless) utility. When the environment becomes a game these individuals have a dominant strategy. The presence of regret neutral individuals makes it more difficult for the regret averse individuals to coordinate on the \emph{other} lottery. The reason is that the regret neutral individuals will always choose one of the lotteries.

The second extension in Appendix \ref{hetero-prefs} considers the more general case where all individuals may have a different coefficient of regret aversion. Ex-ante, the regret coefficients are unknown, with everyone's coefficient being a random draw from some distribution over $[0, \infty)$. This generates a rich Bayesian game. In this sense, the first extension above is is one of many possible realisations of the strategic environment.

Individuals who are not regret averse may be regret neutral or \emph{rejoice lovers}.
Rejoice is defined as the psychological gain that a decision maker -- so called rejoice lover -- experiences when the option chosen turns out to be better than the unchosen option. Appendix \ref{rejoice} describes how rejoice can be incorporated in our model. When a rejoice lover does not learn the lottery's outcome, he will be \emph{more} likely to choose the lottery, as he wants to know whether his choice was the best. It is easy to see that, while in equilibrium regret averse players will coordinate, rejoice lovers will anti-coordinate.

\section{Experiments}\label{experiments}

We test the predictions of a two-player variant of the regret game through two
experiments. Both experiments use a within-subject design and have two main goals. The first goal is identifying the participants who are regret averse. The second goal is testing whether regret averse participants behave as our model predicts. The two experiments achieve these goals through two different designs, with the second experiment serving as a robustness check of the results by eliminating some confounds that may have been present in the first experiment.

\subsection{Experiment 1}\label{experiment1}

\subsubsection{Design}

\paragraph{Overview}
The experiment consists of two parts. In the first part we elicit participants' preferences over a riskless option (a sure amount of money) and a risky option (a lottery) under two different information environments. In the first environment participants learn the risky lottery's outcome even if they do not choose it, while in the second environment they do not learn the risky lottery's outcome unless chosen. These initial decisions allow us to classify participants into regret averse types and non regret averse types and to calibrate the parameters of the second part of the experiment.
We also ask an additional question as a robustness check -- to verify whether participants behave according to the type they were classified as.
In the second part of the experiment, participants are matched in pairs and play the regret game. They must choose between a sure amount of money and a risky lottery. If they do not choose the lottery, they will learn its outcome only if their partner chose the lottery.

\paragraph{Part 1} In the first part of the experiment, Decisions 1 through 3, participants have to choose between a sure amount (\euro{5} with certainty) and a
risky lottery (\euro{}$x$ with 50\% probability and \euro{0}
with 50\% probability) under different conditions.

The standard incentive compatible mechanism for eliciting lottery thresholds is the BDM \citep{BDM}, and by Theorem \ref{thm:regretEquivalent} the BDM remains incentive compatible for regret averse individuals. We ask each participant to state the smallest lottery outcome $x$ (henceforth \emph{lottery threshold}) such that they prefer playing the lottery than receiving the sure
amount. They
can choose any number from the list $\{5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}$. After they submit their choice, the computer randomly picks a number from the same list, independently drawn for
each participant. All the numbers are equally likely. If the number
picked by the computer is smaller than the number $x$ chosen by the participant, the sure amount is the implemented option,
i.e., the participant receives \euro{5}. If the number picked by the
computer is equal or larger than the number $x$ chosen by the participant, the lottery is the implemented option, i.e.,
the participant receives the number picked by the computer in
\euro{} with 50\% probability and \euro{0}
otherwise.\footnote{Whether the lottery is successful or not is perfectly correlated across participants.}

The above is common to Decisions 1 and 2 (and, as discussed later, to Decision 3). The difference between the decisions is seen in the information feedback provided to the participants who do not choose the lottery. Labelling the number $x$ chosen by participants in Decision 1 as $x_1$ and that chosen by participants in Decision 2 as $x_2$, we have the following.

\bigskip

\emph{Decision 1}. If a participant's choice, $x_1$, results in the sure amount being the implemented option, they are nevertheless informed about the lottery's
outcome.

\bigskip

\emph{Decision 2}. If a participant's choice, $x_2$, results in the sure amount being the implemented option, they are \emph{not} informed about the lottery's outcome.

\bigskip

After each of the first two decisions, participants learn the number randomly picked by the computer,
and thus whether the implemented option is the lottery or the sure amount. In the latter case, the information provided varies across the two decisions as described above.

Participants who choose a higher lottery threshold in Decision 2 (no information) than in Decision 1 (information), i.e., $x_2>x_1$, are classified as \emph{regret averse}. The sure amount is more appealing to them when feedback about the lottery is withheld, as it allows them to remain ignorant about the outcome of the unchosen option.
Participants who choose the same or a lower lottery threshold in Decision 2 than in Decision 1, i.e., $x_2\leq x_1$, are classified as \emph{non regret averse}. These participants can be further classified in two categories. Participants who choose $x_2=x_1$ are \emph{regret neutral}, as they do not react to feedback about the unchosen option. Participants who choose $x_2<x_1$ are \emph{rejoice lovers}. The sure amount is less appealing to them when feedback about the lottery is withheld, as they cannot learn the lottery's outcome unless they choose it.\footnote{Since we use a within-subject design, we are potentially exposed to the risk of experimenter demand effect. However, we believe that it is very difficult for participants to figure out what the experimenter is trying to achieve through Decisions 1 and 2, and the heterogeneity in participants' responses seems to support our belief.}

To the best of our knowledge, our paper is the first to classify participants into regret averse and non regret averse types by using an incentive-compatible mechanism under different feedback conditions. However, the idea that the expectation of feedback affects anticipated regret and thus the behaviour of regret averse individuals is has been documented by psychologists. For example, \citet{Zeelenberg:1996} show that, when faced with the choice between two equally attractive gambles, most participants chose the gamble without feedback and thus ex-post comparisons of outcomes.\footnote{For a different method to classify subjects into regret averse and not, see \citet{BleichrodtCillo:2010:MS}. Similarly to our design, \citet{ImasLameWilson} compare participants' valuations for identical lotteries under two different feedback scenarios. However, they use a between-subject design and it is not possible to back out the number of regret averse individuals from their data.}

By classifying participants into types on the basis of a single decision, we are exposed to the risk of measurement error. However, if multiple decisions had been used to classify participants into types, we would have faced two other issues. First, earlier decisions may have affected later decisions, and varying the order to control for any possible order effect may not have been possible. Second, by going through multiple, similar questions participants may have felt bored and answered later question less accurately. In our second experiment, we use an alternative method to classify participants into types. While the new classification is still based on one decision only, by employing a different method it can serve as a robustness check for the classification into types used in our first experiment.

Finally, we ask participants an additional question (Decision 3) to verify whether they behave according to the type they were classified as through Decisions 1 and 2. Each participant is randomly paired with another participant and faces the same decision as in Decisions 1 and 2, with one difference:\ the feedback provided to the participants who does not play the lottery. If the number $x$ chosen by a participant in Decision 3, labelled $x_3$, results in the sure amount being the implemented option, they will be informed about the lottery's outcome only if their partner played the lottery.
This means that if their partner played the lottery,
participants are under the same information environment as in Decision 1, where they learn the
lottery's outcome. If their partner did not play the lottery, they are under the same information environment as in Decision 2, where they do not learn the lottery's outcome unless they play it.

Participants are also asked what they believe is the lottery threshold $x$ chosen by their partner in Decision 3. If their guess is within one of the number chosen by their partner, then they receive additional \euro{1} at the end of the study (provided that Decision 3 is randomly selected for payment).

As with Decisions 1 and 2, in Decision 3 participants are informed as to the number randomly picked by the computer,
and thus whether the implemented option is the lottery or the sure amount.

\paragraph{Part 2}

In the second part of the experiment (Decision 4 onwards), participants play the regret game described in Section \ref{subsec:strategicSetting}. They must choose between a sure amount (earning \euro{5} with certainty) and a lottery (earning ``an amount'' in \euro{} with 50\% probability and \euro{0} with 50\% probability).
Differently from the previous part of the experiment, the lottery's outcome in the good state is given and, for each participant, it is determined using the lottery thresholds $x_1$ and $x_2$ elicited in Decisions 1 and 2.

The sure amount is known regardless of the decision made. If participants choose the sure amount, they learn the lottery's outcome only if their partner chooses the lottery. This implies that if participants choose the lottery, they experience regret if the bad state materialises, whereas if they choose the sure amount, they experience regret if the good state materialises \emph{and} they learn that.

This setting is identical in all the decisions in the second part of the experiment.
However there are three type of decisions that differ in the amount the lottery yields in the good state corresponding to the three parts of the parameter space classified in Theorem \ref{thm:Nashmany}. Two types of decisions are designed to test the regret game when it has a dominant strategy -- in one the dominant strategy is the sure amount and in the other the dominant strategy is the lottery. The third type of decision is designed to test the regret game when it is a game of coordination. More interestingly, in each of these three decisions the amount that the lottery yields in the good state is individual-specific:\ it depends on a participant's choices in Decisions 1 and 2. This ensures that each participant, despite potential heterogeneity in their (regret averse) preferences, plays the regret game in its three key cases.

\bigskip

\emph{Decision 4}. The lottery's outcome in the good state is \emph{smaller} than the amount a participant chose in Decision 1, namely $x_1-2$. This allows us to test the prediction of the regret game when choosing the sure amount is the unique equilibrium in dominant strategies.

\bigskip

\emph{Decision 5}. The lottery's outcome in the good state is \emph{bigger} than the amount a participant chose in Decision 2, namely $x_2+2$. This allows us to test the prediction of the regret game when choosing the lottery is the unique dominant strategy equilibrium.

\bigskip

\emph{Decision 6}. The lottery's outcome in the good state is \emph{in between} the amount chosen in Decision 1 and that chosen in Decision 2, namely $\frac{x_1+x_2}{2}$. This allows us to test the prediction of the regret game when it is a game of coordination.

\bigskip

Decision 6, being the most interesting case of the regret game and thus the core part of the experiment, is repeated 20 times (Decisions 6 to 25).

Pairs are rematched before Decisions 4, 5 and 6. This means that a participant's partner could be the same as before or a different one. To mitigate potential dependence resulting from the repeated interaction of participants, we use matching groups of size 4. That is, for each participant, there are three potential participants who can be randomly assigned to them. In Decision 6 and its repetitions, participants keep the same partner.

From Decision 4 onwards, we elicit first order beliefs. We ask participants to guess whether their partner chose the sure amount of money or the risky lottery. If they guess their partner's choice, they receive additional \euro{1} at the end of the study, if that decision is randomly selected for payment.

After the last iteration of Decision 6, i.e., Decision 25, the participants are asked some non-incentivised questions:\ the Regret Scale \cite{Schwartz:2002:JPSP}, the Big-Five personality traits \cite{Gosling:2003:JRP} and demographic characteristics.

\paragraph{Procedure}
The sessions were run in May 2017 in the experimental laboratory at
the University of Bonn. The experiment was programmed and conducted
with the software z-Tree \cite{Fishbacher:2007:EE}.

At the end of the experiment, each participant received:\ (i) a show-up fee of \euro{4}, (ii) the payment for one randomly chosen decision out of the 25 decisions made, (iii) the payment for one randomly chosen belief out of the 23 beliefs elicited, but only in case of a correct guess. On average, a participant earned \euro{11.50}.

\subsubsection{Testable predictions}\label{predictions}

Decision 6 and its repetitions test Theorem \ref{thm:Nashmany} Part \ref{condition:Nashmany3}, i.e., they test the predictions of the regret game when it is a game of coordination with two pure strategy Nash Equilibria. A regret averse type will choose the lottery if he believes that his partner chose the lottery, and the sure amount otherwise. This yields our first testable prediction.

\begin{pred}\label{pred3} In Decision 6 and its repetitions, believing that his partner will choose the lottery (sure amount) increases a regret averse agent's probability of choosing the lottery (sure amount).
\end{pred}

\bigskip

Decision 4 tests Theorem \ref{thm:Nashmany} Part \ref{condition:Nashmany1}, i.e., it tests the predictions of the regret game when choosing the sure amount is the unique equilibrium in dominant strategies. Decision 5 tests Theorem \ref{thm:Nashmany} Part \ref{condition:Nashmany2}, i.e., it tests the predictions of the regret game when choosing the lottery is the unique equilibrium in dominant strategies. When the regret game has a unique equilibrium in dominant strategies, an agent will choose it regardless of their regret preferences and regardless of their beliefs.
This yields our second testable prediction.

\begin{pred}\label{pred2}  Regret averse agents will choose the sure amount in Decision 4 and the lottery in Decision 5, regardless of their beliefs.
\end{pred}

We also have a third prediction, aimed at testing whether, under a partner-dependent information condition (basically a threshold-based variant of the regret game), participants behave according to the type they were classified as in Decisions 1 and 2.
In Decision 3, regret averse (and rejoice loving) participants should choose the same lottery threshold as in Decision 1 if they believe that their partner played the lottery (i.e., if they expect to learn the lottery's outcome) and the same lottery as in Decision 2 if they believe that their partner did not play the lottery (i.e., if they expect not to learn the lottery's outcome).
That is, they should choose a lottery threshold in between the lottery thresholds chosen under information and under no information.

As $x_2>x_1$ for regret averse types and $x_2<x_1$ for rejoice loving types, for regret averse (rejoice loving) participants $x_3$ should exceed (fall short of) $x_1$ and fall short of (exceed) $x_2$. For regret neutral participants $x_3$ should equal $x_1$ and $x_2$:\ they should choose the same lottery threshold as in Decisions 1 and 2, independently of their beliefs.

\begin{pred}\label{pred1} For regret averse (rejoice loving) agents, $x_3$ will exceed (fall short of) $x_1$ and fall short of (exceed) $x_2$. For regret neutral agents, $x_1=x_2=x_3$.
\end{pred}

\subsubsection{Results}

A total of 144 subjects participated in the experiment with over 90\% of them students and 56\% of them female. The average age was 25 (24 among students and 27 among non students).
In our sample, 22\% of the participants chose $x_2>x_1$ and are classified as regret averse. Half the participants chose $x_2=x_1$ and are classified as regret neutral. The remaining participants chose $x_2<x_1$ and are classified as rejoice loving. Figure \ref{fig:thresholds} shows the distribution of the difference between $x_2$ and $x_1$, which is a proxy for the strength of participants' regret aversion.

\begin{figure}[hhh]
\centering \includegraphics[scale=0.8]{thresholds}
\caption{Difference between $x_2$ and $x_1$.}
\label{fig:thresholds}\end{figure}

To test Prediction \ref{pred3}, we estimate a
logit panel-model with random effects, where the unit of observation is the individual observed in Decisions 6 through 25. We report the marginal effect of an agent's belief that his partner chose the lottery on her own likelihood of choosing the lottery.\footnote{We cluster the standard errors at the matching group-level.}
The dependent variable \emph{lottery choice} takes value 1 if the agent chooses the lottery and 0 if he chooses the sure amount. We use the following independent variables. The variable
\emph{belief} equals 1 if the agent believes that his partner
chose the lottery in the current round and 0 if he believes
that his partner chose the sure amount. The variables \emph{belief if regret averse} and \emph{belief if non regret averse} equal \emph{belief} if, respectively, the agent is regret averse and non regret averse (i.e., regret neutral or rejoice lover).
The variable \emph{past regret} captures regret
generated by previous decisions, and equals 1 (i) if an agent
has not chosen the lottery in the previous round, while his partner
has, and the lottery has been successful, or (ii) if an agent
has chosen the lottery in the previous round and the lottery was not
successful. It equals 0 otherwise.\footnote{Note that, while both
(i) and (ii) can be interpreted as regret driven by past,
unsuccessful decisions, their nature can potentially differ. While
(i) captures peer-induced regret, as well as personal loss, (ii)
only captures loss. Given that, we also repeat our regressions
splitting \emph{past regret} into two dummies respectively
corresponding to cases (i) and (ii). Our results do not change.}
In column (1), we only control for \emph{beliefs if regret averse} and \emph{beliefs if non regret averse}. In column (2), we additionally control for \emph{past regret}.
In column (3), we additionally control for demographics (female dummy, student dummy and age).

We find that, consistently with Prediction \ref{pred3}, believing that their partner chose the lottery significantly increases regret averse participants' likelihood of choosing the lottery. We observe a positive impact of beliefs on choice also for non regret averse participants. However, for non regret averse participants the magnitude of the marginal effects is smaller than for regret averse participants.

\begin{table}[htbp]\centering
\def\sym#1{\ifmmode^{#1}\else\(^{#1}\)\fi}
\caption{Impact of beliefs on choice (all rounds)\label{regret-belief}}
\setlength\tabcolsep{20pt}
\begin{tabular}{l*{3}{c}}
\hline
 DV: lottery choice          &\multicolumn{1}{c}{(1)}         &\multicolumn{1}{c}{(2)}         &\multicolumn{1}{c}{(3)}  \\
         \hline
 & & &\\
belief if regret averse &       0.175\sym{***}&       0.167\sym{***}&       0.166\sym{***}\\
        &     (0.032)         &     (0.033)         &     (0.033)    \\
belief if non regret averse    &       0.172\sym{***}&       0.159\sym{***}&       0.157\sym{***}\\
            &     (0.022)         &     (0.022)         &     (0.022)     \\
past regret    &       No &       Yes &       Yes\\
female dummy            &    No         &    No       &   Yes     \\
student dummy            &    No         &    No       &   Yes     \\
age            &    No         &    No       &   Yes     \\
\hline
N       &        2880         &        2736         &        2736         \\
\hline
\end{tabular}
\begin{tablenotes}
\item {\footnotesize Marginal effects from logit regression. \sym{*} \(p<0.1\), \sym{**} \(p<0.05\), \sym{***} \(p<0.01\). Standard errors in parentheses, clustered at matching group-level. DV equals 1 if the agent chose the lottery and 0 otherwise.}
\end{tablenotes}
\end{table}

Interestingly, when we focus on the last iteration of Decision 6 (Table \ref{last_round}), where we expect that learning may have helped participants to converge to equilibrium, we observe that the impact of beliefs on choice is larger than in Table \ref{regret-belief} and highly significant for regret averse participants. In contrast, it is only marginally significant for non regret averse participants. Moreover, the difference in the magnitude of the marginal effects between regret averse and non regret averse is larger in the last iteration than in all rounds pooled. Our first and core result follows.

\begin{table}[htbp]\centering
\def\sym#1{\ifmmode^{#1}\else\(^{#1}\)\fi}
\caption{Impact of beliefs on choice (last round)\label{last_round}}
\setlength\tabcolsep{20pt}
\begin{tabular}{l*{3}{c}}
\hline
 DV: lottery choice         &\multicolumn{1}{c}{(1)}         &\multicolumn{1}{c}{(2)}         &\multicolumn{1}{c}{(3)}  \\
          \hline
 & & &\\
belief if regret averse &       $0.301\sym{***}$ &       $0.291\sym{***}$ &       $0.284\sym{***}$ \\
        &     (0.091)         &     (0.092)         &     (0.093)         \\
belief if non regret averse    &       $0.195\sym{**}$ &       $0.179^{**}$ &       $0.162\sym{*}$ \\
           &     (0.085)         &     (0.085)         &     (0.087)            \\
past regret    &       No &       Yes &       Yes\\
female dummy            &    No         &    No       &   Yes     \\
student dummy            &    No         &    No       &   Yes     \\
age            &    No         &    No       &   Yes     \\
\hline
N       &        144         &        144         &        144          \\
\hline
\end{tabular}
\begin{tablenotes}
\item {\footnotesize Marginal effects from logit regression. \sym{*} \(p<0.1\), \sym{**} \(p<0.05\), \sym{***} \(p<0.01\). Standard errors in parentheses, clustered at matching group-level. DV equals 1 if the agent chose the lottery and 0 otherwise.}
\end{tablenotes}
\end{table}

\begin{result} In Decision 6 and its repetitions, believing that their partner chose the lottery (sure amount) significantly increases regret averse participants' probability to choose the lottery (sure amount). This effect becomes stronger in the last iteration of the game.
\end{result}

This result indicates that regret aversion drives coordination.
However, in our setting there may have been two additional drivers of coordination:\ preferences for conformism \cite{CharnessRigottiRustichini:2017:GEB,CharnessNaefSontuoso:2019:JET} and inequity averse preferences \cite{FehrSchmidt:1999:QJE}.
As further discussed in Section \ref{experiment2}, we ran a second experiment to eliminate these potential confounds.

Table \ref{prediction2} reports the percentages of participants choosing the lottery in Decisions 4 and 5, i.e., when the regret game has a dominant strategy. Consistently with Prediction \ref{pred2}, the large majority of regret averse participants chose the dominant strategy, i.e., the sure amount in Decision 4 and the lottery in Decision 5. As expected, also the large majority of non regret averse participants followed this pattern of decisions.\footnote{In Table \ref{prediction2} the non regret averse participants  include regret neutral participants, and rejoice lovers when the game had a dominant strategy ($x_1-x_2\leq 2$). As expected, when the game had no dominant strategy ($x_1-x_2> 2$), the percentages of rejoice lovers choosing the sure amount in Decision 4 and the lottery in Decision 5 were remarkably lower.}

\begin{table}\centering
\def\sym#1{\ifmmode^{#1}\else\(^{#1}\)\fi}
\caption{Participants choosing the lottery in Decisions 4 and 5}\label{prediction2}
\centerline{
{
\begin{tabular}{c|c|c}
\hline
                    & \textbf{regret averse}  & \textbf{non regret averse}\\
\hline
Agents who chose lottery in D4         &    $6\%$   & $20\%$ \\
Agents who chose lottery in D5        &       $66\%$ &   $80\%$ \\
\hline
\end{tabular}
}
}
\end{table}

\bigskip

\begin{result} Most regret averse agents choose the sure amount in Decision 4 and the lottery in Decision 5.
\end{result}

Finally, as a robustness check, we verify whether under a partner-dependent information condition (Decision 3), participants behave consistently with the type they were classified as through Decisions 1 and 2. Table \ref{robustness} presents the amounts chosen in Decision 1, Decision 2 and
Decision 3 -- overall and broken down by type. We find that, in line with Prediction \ref{pred1}, for regret averse participants the mean lottery threshold chosen in Decision 3, $x_3$, is higher than mean $x_1$ and lower than mean $x_2$. For rejoice loving participants, the opposite happens, and for regret neutral participants mean $x_3$ does not significantly differ from mean $x_1$ and mean $x_2$.

\begin{table}\centering
\def\sym#1{\ifmmode^{#1}\else\(^{#1}\)\fi}
\caption{Mean lottery thresholds chosen in Decisions 1-3 by type}\label{robustness}
\setlength\tabcolsep{10pt}
\centerline{
{
\begin{tabular}{lccc|c}

\hline
                    & \textbf{regret averse}  & \textbf{regret neutral}  &  \textbf{rejoice lover} & \textbf{all}   \\
\hline
Decision 1 ($x_1$)             &   9.15  &   11.53  &    12.33 & 11.22\\
              &       (2.40)&   (2.04) &    (1.98) &(2.39)\\
Decision 2 ($x_2$)         &    12    & 11.10    & 9.77   & 11.16 \\
              &       (2.56)&   (2.70) &    (2.34) &(2.40)\\
Decision 3 ($x_3$)         &       11.25 &    11.64 &    11.10 & 11.41\\
              &       (2.59)&   (2.30) &    (2.70) &(2.47)\\
  \hline
$x_1-x_3$ & -2.1\sym{***} & -0.11 & 1.23\sym{**} & -0.19\\
$x_2-x_3$ & 0.75\sym{*} & -0.54 & -1.33\sym{*} & -0.25\\
\hline
N        & 32  & 73  & 39 & 144  \\
\hline
\end{tabular}
}
}
\begin{tablenotes}
\small\item {\footnotesize Standard deviation in parentheses. The Wilcoxon test tests $H_0 : x_1-x_3=0$ and $H_0 : x_2-x_3=0$. \sym{*} \(p<0.1\), \sym{**} \(p<0.05\), \sym{***} \(p<0.01\).}
\end{tablenotes}
\end{table}

To check whether these differences are statistically significant, we run the Wilcoxon equality test on matched data. The null hypotheses $H_0 : x_1-x_3=0$ and $H_0 : x_2-x_3=0$ are rejected for regret averse participants and rejoice loving participants, and not rejected for regret neutral participants. These results strongly support Prediction \ref{pred1}, thereby offering some reassurance that participants have been classified into types accurately.

\begin{result} The partner-dependent lottery threshold $x_3$ is significantly higher (lower) than the lottery threshold $x_1$ and significantly lower (higher) than the lottery threshold $x_2$ for regret averse (rejoice loving) participants. It is not significantly higher than $x_1$ and $x_2$ for regret neutral participants.
\end{result}

Prediction \ref{pred1} implies that in Decision 3, the correlation between lottery threshold chosen and belief about the partner's lottery threshold, denoted by $\chi$, will be higher for regret averse participants than for non regret averse participants, i.e., $\chi_{R}>\chi_{NR}$.
This is due to the fact that regret aversion induces a desire to coordinate. We find that $\chi_{R}$ equals 0.77 and $\chi_{NR}$ equals 0.47.
The test for equality of correlation
coefficients rejects the null hypothesis $H_0 : \chi_{R}=\chi_{NR}$. In particular, $\chi_{R}$
is significantly higher than $\chi_{NR}$ ($p = 0.01$).

\subsection{Experiment 2}\label{experiment2}

\subsubsection{Design}

\paragraph{Overview} If participants have preferences for conformism \cite{CharnessRigottiRustichini:2017:GEB,CharnessNaefSontuoso:2019:JET}, they may have chosen what they believed their partner to choose because in the game in Experiment 1 they did not know ex ante what the best choice was and thus preferred to do what their partner did in a previous round or what they believed their partner did in the current round. Moreover, if participants have inequity averse preferences \cite{FehrSchmidt:1999:QJE}, they may have coordinated on the decision of their partner because in the game in Experiment 1 their earnings may have substantially differed from their partner's earnings and they did not want to risk to earn less than their partner.
To eliminate these potential confounds we designed and ran a second experiment.

The main goal of Experiment 2 is to test our model's key prediction (i.e., that regret averse players try to coordinate with their partner) using a one shot variant of the regret game that eliminates the aforementioned potential confounds. The secondary goal of Experiment 2 is to provide an alternative and simple method to classify participants into regret averse and non regret averse types, which serves as a robustness check for the classification into types provided by Experiment 1.

Experiment 2 consists of two parts. In the first part, participants  have to choose between a sure amount of money and a lottery. This decision is calibrated such that they prefer the sure amount. Then they are asked whether they want \emph{to find out} how much they would have earned had they chosen the risky lottery.
If they choose \emph{not to find out}, they forgo a small amount of money. This question allows us to classify participants into regret averse types and non regret averse types. In the second part of the experiment, participants are matched in pairs and play a variant of the regret game: they must choose whether they want to find out the risky lottery's outcome or not, but they can avoid finding out only if their partner decides not to find out too.

\paragraph{Part 1} In this part subjects have to take 3 decisions.

\emph{Decision 1.}  We elicit each participant's valuation of a lottery. We ask each participant for the smallest sure amount of money that they would choose over a risky lottery paying \textsterling80 with 20\% probability and \textsterling0 with 20\% probability.\footnote{To make the lottery easier to understand, we told them that the computer would randomly draw a ball from an urn containing 4 blue balls and 1 red ball. If the ball drawn was blue, they would earn \textsterling0; if the ball drawn was red, they would earn \textsterling80. We also showed a picture of the urn. See Appendix \ref{instructions2} for further details.} We use an incentive compatible mechanism, the BDM, in the simpler version developed by \citet{healy}. Participants are shown a list of 80 questions, each asking if they prefer the risky lottery (referred to as Option A) or a sure amount of money (referred to as Option B). The sure amount of money ranges from \textsterling1 (first question) to \textsterling80 (last question). Participants are then asked at which value of Option B they want to switch from Option A to Option B. Only 10\% of the participants are paid for this decision.\footnote{To ensure that only participants who read the instructions carefully remain in the study, participants who chose a number higher than twice the expected value of the lottery (24\% of the initial sample) ended the experiment after this decision.}

\emph{Decision 2.} We ask participants to choose between a sure amount of money (equal to the amount elicited in Decision 1 plus \textsterling2) and the risky lottery paying \textsterling80 with 20\% probability and \textsterling0 with 80\% probability. In order to be consistent with Decision 1, participants should choose the sure amount of money. All participants are paid for their decision in Decision 2.

\emph{Decision 3}. All the participants who choose the sure amount in Decision 2 are asked whether they want to find out how much they would have earned, had they chosen the risky lottery.\footnote{The participants who chose the lottery in Decision 2 (9\%), thereby contradicting their previous decision, were then asked alternative questions and excluded from our data analysis. For further details, see Appendix \ref{instructions2}.} By choosing not to find out, they forgo a small amount of money. In particular, if they chose to find out, they are informed about the lottery's outcome at the end of the experiment, and their earnings are increased by \textsterling0.04. If they chose not to find out, they are not informed about the lottery's outcome at the end of the experiment, and their current earnings are not increased by \textsterling0.04.
Those who choose not to find out are classified as regret averse and those who chose to find out as non regret averse.

\paragraph{Part 2} In this part subjects play one shot of a variant of the regret game.

\emph{Decision 4.} Participants are matched in pairs. Again, they must choose whether they want to find out the risky lottery's outcome or not. However, whether they learn the lottery's outcome also depends on their partner's behaviour. If they choose not to find out, they can avoid learning the lottery's outcome only if their partner also chose not to find out. Either this decision or the previous decision is randomly selected and implemented. If a participant chooses to find out in the randomly selected decision, their total earnings are increased by \textsterling0.04.

It is now easy to see how the design of Experiment 2 removes the potential confounds generated  by preferences for conformism and inequity averse preferences. First, we use a one shot game in Decision 4, thus participants cannot imitate their partner's decisions in previous rounds. Second, in the variant of the game in Decision 4 participants can immediately identify the best decision to take given their preferences. This eliminates the concern generated by conformism motives, which would be at play if participants were unsure of the best thing to choose. Finally, in Decision 4 the potential earning difference generated by a participant's decision is negligible (at most \textsterling0.04). This eliminates the concern generated by inequity averse motives.

After making their decision, participants are also asked to guess their partner's decision. If they guess their partner's decision, they earn additional \textsterling0.50. Only one of the last two questions is randomly selected and implemented.

Differently from Experiment 1, in Experiment 2 the strategic decision aimed at testing the predictions of the regret game was not repeated. The reasons are the following. First, we wanted to avoid repeated-game effects and particularly the effect of conforming with the partner's previous decisions. Second, given that the strategic decision was simple and built on the previous decision, it did not appear necessary to provide participants with learning opportunities. Third, we thought that the effect of regret on behaviour may be more salient when the game is played only once.

\paragraph{Procedure} Due to COVID 19-related lab closures, the sessions were run online in May 2021. To increase the robustness and external validity of our results, we used two different samples:\ students from Royal Holloway University of London and Prolific participants.
The experiment was programmed and conducted
with the software o-Tree \cite{ChenSchongerWickens}.

At the end of the experiment, 10\% of participants were paid for Part 1 and every participant was paid for Decision 2. If in the randomly selected decision out of Decision 3 and Decision 4, a participant chose to find out the lottery's outcome, their total earnings were increased by \textsterling0.04. If they guessed their partner's decision, they earned additional \textsterling0.50. On average, a participant earned \textsterling18.21.

\subsubsection{Testable predictions}

Decision 4 tests the prediction of the regret game when it is a game of coordination. In Decison 4, a regret averse participant will choose to find out the lottery's outcome if he believes that his partner chose to find out the lottery's outcome too. He will choose not to find out the lottery's outcome if he believes that his partner also chose not to find out the lottery's outcome. This implies that the share of regret averse agents choosing to find out under the belief that their partner chose to find out will be significantly higher than the share of regret averse agents choosing to find out under the belief that their partner chose not to find out. Similarly, the share of regret averse agents choosing not to find out under the belief that their partner also chose not to find out will be significantly higher than the share of regret averse agents choosing not to find out under the belief that their partner chose to find out.
This yields our testable prediction.

\begin{pred}\label{pred4} The fraction of regret averse agents choosing the option that they believe their partner chose will be significantly higher than the fraction choosing the alternative option.
\end{pred}

\subsubsection{Results}

We have a sample of 213 participants who completed the experiment: 84 students from Royal Holloway University and 129 participants from Prolific. 54\% of the participants were female, 44\% were male and the remaining 2\% classified themselves as ``other''. The gender distribution is very similar across our two subsamples. The average age was 27 (21 among Royal Holloway students and 32 among Prolific participants).

In our sample, 12\% of the participants chose not to find out the lottery's outcome and are classified as \emph{regret averse} and 88\% of the participants chose to find out the lottery's outcome and are classified as \emph{non regret averse}.

Table \ref{tab_q3_ra} shows the distribution of choices -- between finding out and not finding out -- and beliefs for regret averse participants. We can observe that 80\% of the regret averse participants chose not to find out, as in the individual decision, and 20\% chose to find out. It makes sense that either option gets chosen, as in Decision 4 there is no dominant strategy. All the participants who expect their partner to choose not to find out, also chose not to find out. Some participants chose not to find out even if they believed that their partner found out. This may have the following explanation. Given that we elicited point beliefs, participants who reported that their partner chose to find out could still believe that with some probability their partner chose not to find out. In this case it would be optimal not to find out, because the small amount of money to forgo is compensated by the reduction of the expected regret.

\begin{table}\centering
\def\sym#1{\ifmmode^{#1}\else\(^{#1}\)\fi}
\caption{regret averse subjects' choices and beliefs in Decision 4}\label{tab_q3_ra}
\setlength\tabcolsep{10pt}
\centerline{
{
\begin{tabular}{c|cc|c}
\hline
                    & Believes partner   & Believes partner   & Total   \\
                    &  finds out  &  does not find out  &   \\
\hline
Finds out             &   5  &    0 & 5\\
Does not find out           & 8    & 12   & 20 \\
  \hline
Total        & 13  & 12  & 25  \\
\hline
\end{tabular}
}
}
\end{table}

Table \ref{tab_q3_nra} shows the distribution of choices -- between finding out and not finding out -- and beliefs for non regret averse participants. Over 98\% of the non regret averse participants (185 out of 188) chose to find out, as in the individual decision. This is expected, as finding out is a dominant strategy for them. It also shows that they are consistent across decisions. Out of these 185 participants choosing to find out, 10 believed that their partner chose not to find out. This is interesting, as it further confirms that finding out is a dominant strategy for them.

\begin{table}\centering
\def\sym#1{\ifmmode^{#1}\else\(^{#1}\)\fi}
\caption{Non regret averse subjects' choices and beliefs in Decision 4}\label{tab_q3_nra}
\setlength\tabcolsep{10pt}
\centerline{
{
\begin{tabular}{c|cc|c}
\hline
                    & Believes partner   & Believes partner   & Total   \\
                    &  finds out  &  does not find out  &   \\
\hline
Finds out             &   175  &    10 & 185\\
Does not find out           & 0    & 3   & 3 \\
  \hline
Total        & 175  & 13  & 188  \\
\hline
\end{tabular}
}
}
\end{table}

To check whether the differences observed in Table \ref{tab_q3_ra} are statistically significant, we run a t-test. Our null hypothesis is that the frequency with which that a regret averse participant chooses to find out under the belief that his partner did the same equals the frequency with which he chooses to find out under the belief that his partner chose not to find out. The relative frequencies are 1 and 0.61, respectively. We reject the null hypothesis (p=0.0151). Our results support Prediction \ref{pred4}.

\begin{result} The frequency with which regret averse participants choose the option that they believe their partner chose is significantly higher than the frequency with which they choose the alternative option.
\end{result}

\section{Conclusion}\label{CONCLUSION}

This paper began with the following simple observation:\ in many situations, ranging from technology adoption to ordering food in a restaurant, learning the outcome of unchosen alternatives is not guaranteed. Given that a decision maker can only regret a foregone alternative if she learns its outcome, what should be done? We showed how to incorporate this observation into the classic model of a decision maker who is regret averse. Our first contribution is a formalisation of ex post information structures that allow for the possibility that unchosen alternatives are / are not learned about.  That is, the domain of preferences needs to be extended from simply `objects of choice' to `objects of choice and their associated information environment'. For a given choice set, we provide a definition that ranks two informational environments according to which is ``more informative''. And we show, in Theorem \ref{thm:lessInfo}, that a more informative environment is never preferred for a regret-averse decision maker.

In Section \ref{subsec:strategicSetting}, we allow for the possibility that the ex-post information environment that will be faced depends not only on one's own choice but also on the choices of others. Thus, what on the surface appears to be a collection of individual decision problems - like for example ordering food in a restaurant - is in fact a rich multi-player behavioural game. The reason, of course, is that the decisions of others - one's fellow diners - can be informative about foregone alternatives, and for a regret averse individual that matters. We term this environment the regret game, and in Theorem \ref{thm:Nashmany} we classify conditions on preferences for which it is a coordination game with multiple equilibria.

We tested the predictions of our model through two experiments. Both experiments have two main goals:\ identifying the participants who are regret averse and testing whether they behave as our theory predicts. In the first experiment, we find that, as predicted by our model, regret averse participants try to coordinate with their partner. Believing that their partner chose an option significantly increases their likelihood of choosing that option. We observe a positive impact of beliefs on choice also for non regret averse participants. However, for non regret averse participants this impact is smaller. Moreover, when we focus on the last iteration of the game, the impact of beliefs on choice is larger and highly significant for regret averse participants, but is only marginally significant for non regret averse participants.

These results indicate that regret aversion drives coordination.
However,  preferences for conformism \cite{CharnessRigottiRustichini:2017:GEB,CharnessNaefSontuoso:2019:JET} and inequity averse preferences \cite{FehrSchmidt:1999:QJE} may have been two additional drivers of coordination. We ran a second experiment to eliminate these potential confounds.

The results of the second experiment support the key findings of the first experiment:\ regret averse participants try to coordinate with their partner. The probability that they choose what they think their partner chose is significantly higher than the probability that they choose the alternative option. Most notably, all the regret averse participants who believe that their partner chose to avoid information, choose to avoid information too. Instead, almost all of the non regret averse participants (over 98\%) play the dominant strategy, and some of them do so under the belief that their partner chose the alternative option. Thus, once the aforementioned confounds are removed by design, we still find strong support for our model's predictions.

It is traditional in economic modelling to view the information available to individuals as part of the environment and not something over which there is choice. While we have focused on regret, one can imagine a host of other behavioural explanations for why an individual might (i) want to avoid information, and then (ii) take steps in order to avoid information. It then immediately follows, that since the behaviour of others can impact one's available information, there are strategic interdependencies at play. It is our hope that more models of the sort presented here can be developed to better-understand the role that evolving information plays for economic agents, and in particular the way in which it connects them.

\newpage

\bibliographystyle{elsarticle-harv}