2.1 Model

We consider a framework in which conflict about a prize takes place in up to n stages, each described as a Tullock contest. The players differ in their prize valuation and there is uncertainty about the probability distribution of types. The theory framework allows for learning about the true type distribution but, by construction, removes strategic aspects of how own effort choices are interpreted by others. One variant of the analysis allows for players’ selection by including the possibility to exit the multi-stage game.

Players, actions, and timing Let I be an infinitely large set of players. The game has up to n stages but may end before reaching the terminal stage. In any given non-terminal stage $s<n$ , if this stage is reached, each player i is randomly matched with one other player $-i$ . Players i and $-i$ simultaneously choose efforts $x_{i,s}\in [\underline{x},\overline{x}]$ and $x_{-i,s}\in [\underline{x},\overline{x}]$ , where $0<\underline{x}<\overline{x}$ . This leads to one of three outcomes, described from the point of view of player i. In the first outcome i wins a prize and reaches no further stage (that is, the game ends for i). In the second outcome player $-i$ wins this prize; again, i reaches no further stage (the game ends for i). In the third outcome none of the two players wins the prize but the game continues for i who enters stage $s+1$ . This third outcome emerges with probability $1-q$ . This probability is exogenously given and does not depend on $x_{i,s}$ or $x_{-i,s}$ . The other two outcomes emerge with probabilities $q{p}_{i,s}(x_{i,s},x_{-i,s})$ and $q(1-p_{i,s}(x_{i,s},x_{-i,s}))$ . As will become clear below, the assumption of an exogenous continuation probability $1-q$ is the key assumption that allows to isolate selection effects in the experimental data.

The function $p_{i,s}(x_{i,s},x_{-i,s})$ describes the win probability of player i in stage s, conditional on the prize being awarded in this stage. This conditional probability is a function of the player’s own effort and the opponent $-i$ ’s effort at this stage. We assume that this function is given by the Tullock (Reference Tullock and Buchanan1980) contest success function:

(1) $\begin{array}{c}{p}_{i,s}(x_{i,s},x_{-i,s})=\frac{x_{i,s}}{x_{i,s}+x_{-i,s}}\end{array}$

for all stages $s=1,\dots ,n$ .Footnote ¹¹ The function $p_{i,s}(x_{i,s},x_{-i,s})$ is continuous, strictly increasing and concave in player i’s own effort and strictly decreasing in the effort of the opponent $-i$ of this stage.Footnote ¹²

If the prize is not allocated in stage $s<n$ such that player i enters into stage $s+1$ , the players are randomly re-matched. Hence, the identity of the opponent typically changes between stages, as the set I of players is infinitely large. We denote by i a given player (with unchanged identity over all stages that are reached) and by $-i$ the opponent assigned to player i in a given stage. In stage $s+1$ , player i and the new opponent $-i$ choose efforts $x_{i,s+1}\in [\underline{x},\overline{x}]$ and $x_{-i,s+1}\in [\underline{x},\overline{x}]$ and the stage contest resolves according to the same rules as in stage s. This continues until the game ends for i because one of the players wins the prize, or until the terminal stage $s=n$ is reached. Interaction at the terminal stage n follows the same rules as in previous stages, with one difference: should none of the two players win at stage n , the game ends and no prize is awarded.

Payoffs Payoffs consist of the prize value if the player wins, minus the own effort costs that are ‘all pay’: Players i and $-i$ pay the cost of their own efforts $x_{i,s}$ and $x_{-i,s}$ . By normalization, these costs are equal to $x_{i,s}$ and $x_{-i,s}$ . They occur independent of whether or not a player wins at stage s, or whether the prize is awarded in this stage at all. There is no discounting, so effort costs add up for the different stages.

Player i values winning the prize by $v_i>0$ , which is private information. This fact and the probability model that describes the random process behind the assignment of valuations is common knowledge and formalized below. Player i learns her own prize valuation $v_i$ prior to the first effort choice in stage 1. We assume that a player keeps this valuation of winning throughout all stages of the game. Players may differ in their valuation of the prize: one share of players has a valuation $v_L$ , the other share of players has a valuation $v_H>v_L$ .Footnote ¹³ We sometimes call a player with valuation $v_L$ a ‘weak’ player and a player with valuation $v_H$ a ‘strong’ player. As there is no discounting, a player has the same benefit from winning the prize if she wins at stage s as if she wins at stage $s+k$ .

Altogether, player i’s payoff is $v_i-{\Sigma }_{k=1}^{k=s}{x}_{i,k}$ if i wins the prize at stage s, $-{\Sigma }_{k=1}^{k=s}{x}_{i,k}$ if $-i$ wins the prize at stage s, and $-{\Sigma }_{k=1}^{k=n}{x}_{i,k}$ if the prize is not allocated at any stage $s=1,\dots ,n$ .

Population uncertainty and information structure The prize valuation $v_i$ is assigned to player i in a random process that has two layers of randomness. First, there are two possible states of the world that may prevail. These states $\overline{\omega }$ and $\underline{\omega }$ differ in the probability distribution from which the players’ valuations are drawn. All players start with common prior beliefs ${\sigma }_i\left(\omega \right)$ about the probability that the world is in state $\omega \in \{\overline{\omega },\underline{\omega }\}$ and attach a probability of 1 / 2 to each of the two possible states.

Second, nature draws the type of each player $i\in I$ independently from the same given probability distribution, which depends on the state of the world. Specifically, in state $\omega$ of the world, player i is assigned valuation $v_H$ with probability ${\pi }_{\omega }$ , and is assigned valuation $v_L$ with the complementary probability $1-{\pi }_{\omega }$ ; we assume that

(2) $\begin{array}{c}{\pi }_{\overline{\omega }}=\frac{1}{2}+d\text{and}{\pi }_{\underline{\omega }}=\frac{1}{2}-d\text{, where}d\in \left(0,,,\frac{1}{2}\right).\end{array}$

Hence, the state of the world characterizes the share of high types, ${\pi }_{\omega }$ , and the share of low types, $1-{\pi }_{\omega }$ , in the population; the share of high types is larger in state $\overline{\omega }$ than in state $\underline{\omega }$ . For $d>0$ the player’s own type as well as the experienced opponents’ efforts affect the players’ updating about the probability for the world to be in state $\overline{\omega }$ or $\underline{\omega }$ .

Beliefs At the beginning of stage 1, each player $i\in I$ learns her valuation $v_i$ , which i keeps throughout the game. As $v_i$ is a random draw from the true probability distribution (which is one of two possible ones), this makes i’s own valuation not only important for the payoff from winning but also for the beliefs about other players’ valuations: player i uses Bayes’ rule to update her belief about the true state of the world, which is then used to determine her beliefs about the composition of the population from which the opponent $-i$ is drawn. In stages $s\ge 2$ (if reached), the beliefs also depend on the history of opponents’ efforts in previous stages $1,\dots ,s-1$ .

Formally, in stage s the population is composed of players with different prize valuations $v_i$ and different histories of observed efforts of previous opponents $-i$ ; the vector

(3) $\begin{array}{c}{h}_{i,s}\equiv (v_i,x_{-i,1},\dots ,x_{-i,s-1})\end{array}$

describes all relevant information about a player’s genuine type (prize valuation) and experience type (history of opponents’ effort choices) at the beginning of stage s. Somewhat loosely we refer to $h_{i,s}\in H_s$ as i’s ‘type’ in stage s where $H_s$ denotes the set of types in stage s. For a player i of type $h_{i,s}$ to be teamed up with player j of type $h_{j,s}$ , the probability beliefs are characterized by cumulative distribution functions $F_{h_{i,s}}\left(h_{j,s}\right)$ . Atoms in these distributions will be denoted by ${\rho }_{h_{i,s}}\left(h_{j,s}\right)$ ; they measure the probability which a player i with valuation $v_i$ and experienced opponents’ effort $\left(x_{-i,1},,,\dots ,,,x_{-i,s-1}\right)$ attributes to the event that her newly matched opponent $-i=j$ in stage s has a prize valuation $v_j$ and experienced previous opponents with expenditures $\left(x_{-j,1},,,\dots ,,,x_{-j,s-1}\right)$ .

2.2 Benchmark: no uncertainty about the state of the world

Proposition 1

Suppose that there is common knowledge about the share ${\pi }_{\omega }$ of high-valuation players and denote the equilibrium efforts by ${\breve{x}}_{v_H}$ and ${\breve{x}}_{v_L}$ for players with valuation $v_H$ and $v_L$ , respectively. Then, ${\breve{x}}_{v_H}$ and ${\breve{x}}_{v_L}$ are constant across all stages s that are reached. All player types expect that their rival’s effort is, on average,

(4) $\begin{array}{c}{E}_{v_H}\left(x_{-i,s}\right)=E_{v_L}\left(x_{-i,s}\right)={\pi }_{\omega }{\breve{x}}_{v_H}+(1-{\pi }_{\omega }){\breve{x}}_{v_L}\text{.}\end{array}$

The proofs of this and all subsequent propositions are in Appendix 1. In the benchmark of a known distribution of types, a player cannot learn from her own type or other players’ effort about future opponents’ types. Thus, each of the stages can be seen as independent; the dynamic game can be interpreted as a sequence of completely independent static games.Footnote ¹⁴ An interior equilibrium $({\breve{x}}_{v_H},{\breve{x}}_{v_L})\in$ ${\left(\underline{x},,,\overline{x}\right)}^2$ at a given stage is described by the first-order conditions

(5) $\begin{array}{c}{\breve{x}}_{v_H}={\pi }_{\omega }\frac{q{v}_H}{4}+\left(1,-,{\pi }_{\omega }\right)\frac{{\breve{x}}_{v_L}{\breve{x}}_{v_H}}{{\left({\breve{x}}_{v_H},+,{\breve{x}}_{v_L}\right)}^2}q{v}_H\end{array}$

and

(6) $\begin{array}{c}{\breve{x}}_{v_L}={\pi }_{\omega }\frac{{\breve{x}}_{v_H}{\breve{x}}_{v_L}}{{\left({\breve{x}}_{v_L},+,{\breve{x}}_{v_H}\right)}^2}q{v}_L+\left(1,-,{\pi }_{\omega }\right)\frac{q{v}_L}{4}.\end{array}$

The equilibrium levels of a player are precisely the same in each stage s and the players’ expectations of the rival’s effort are independent of the own player type.

Special parametric cases have been solved explicitly in the literature. If ${\pi }_{\omega }=1/2$ , the first-order conditions become

$\begin{array}{c}{\breve{x}}_{v_i}=\left(\frac{1}{8},+,\frac{{\breve{x}}_{v_H}{\breve{x}}_{v_L}}{2{\left({\breve{x}}_{v_L},+,{\breve{x}}_{v_H}\right)}^2}\right)q{v}_i\text{for}i\in \{H,L\}\end{array}$

[as in Malueg and Yates (Reference Malueg and Yates2004)]. For the case of symmetric players with $v_H=v_L=v$ , this solution reduces to ${\breve{x}}_v=qv/4$ , which corresponds to the result obtained by Tullock (Reference Tullock and Buchanan1980).

2.3 Perfect Bayesian equilibrium with learning

Lemma 1

In stage $s=1$ , players with valuation $v_H$ and $v_L$ , respectively, believe that the share of high-valuation players in the population is given by

(7) $\begin{array}{c}{\rho }_{v_H}\left(v_H\right)=\frac{1}{2}+2d^2\text{and}{\rho }_{v_L}\left(v_H\right)=\frac{1}{2}-2d^2.\end{array}$

Proposition 3

Denote the equilibrium efforts in stage 1 by $x_{v_H}^{\ast }$ and $x_{v_L}^{\ast }$ for players with valuations $v_H$ and $v_L$ . If $(x_{v_H}^{\ast },x_{v_L}^{\ast })\in$ ${\left(\underline{x},,,\overline{x}\right)}^2$ , these efforts are equal to

(8) $\begin{array}{c}{x}_{v_H}^{\ast }={\rho }_{v_H}\left(v_H\right)\frac{q{v}_H}{4}+\left(1,-,{\rho }_{v_H},\left(v_H\right)\right)\frac{q{v}_H^2{v}_L}{{\left(v_H,+,v_L\right)}^2}\end{array}$

and

(9) $\begin{array}{c}{x}_{v_L}^{\ast }={\rho }_{v_L}\left(v_H\right)\frac{q{v}_H{v}_L^2}{{\left(v_H,+,v_L\right)}^2}+\left(1,-,{\rho }_{v_L},\left(v_H\right)\right)\frac{q{v}_L}{4}.\end{array}$

Equilibrium beliefs about the opponent’s expected effort are

(10) $\begin{array}{c}{E}_{v_H}\left(x_{-i,1}\right)={\rho }_{v_H}\left(v_H\right)x_{v_H}^{\ast }+\left(1,-,{\rho }_{v_H},\left(v_H\right)\right)x_{v_L}^{\ast }\end{array}$

and

(11) $\begin{array}{c}{E}_{v_L}\left(x_{-i,1}\right)={\rho }_{v_L}\left(v_H\right)x_{v_H}^{\ast }+\left(1,-,{\rho }_{v_L},\left(v_H\right)\right)x_{v_L}^{\ast }\end{array}$

with

(12) $\begin{array}{c}{E}_{v_H}\left(x_{-i,1}\right)>E_{v_L}\left(x_{-i,1}\right).\end{array}$

Later stages In stages $s\ge 2$ , player i’s ‘type’ is characterized by the own prize valuation and a history of encounters with other players $-i$ with their own valuation $v_{-i}$ and history of previously matched players. If $H_s$ contained already m different player types, then any player can be matched with any of these types, such that the set $H_{s+1}$ contains $m^2$ elements. In Section B.1 of the Online Appendix we consider properties of stage $s=2$ and establish a ranking of equilibrium efforts (Proposition 6) that demonstrates the potentially countervailing effects of valuation type and experience type on incentives to exert effort. Proposition 6 in the Online Appendix also shows that the stage 2 equilibrium beliefs about the opponent’s effort satisfy

(13) $\begin{array}{c}{E}_{(v_H,x_{v_H}^{\ast })}\left(x_{-i,2}\right)>E_{(v_H,x_{v_L}^{\ast })}\left(x_{-i,2}\right)=E_{(v_L,x_{v_H}^{\ast })}\left(x_{-i,2}\right)>E_{(v_L,x_{v_L}^{\ast })}\left(x_{-i,2}\right)\end{array}$

where, in the subscript, the first element refers to the player’s own valuation and the second element refers to the effort of the stage 1 opponent. Hence, a player’s expectation of the opponent’s effort is still correlated with the own valuation type so that high-valuation players expect, on average, higher opponent’s effort than low-valuation players.

It is evident that calculating the equilibrium efforts for this problem becomes increasingly intractable in later stages. However, we can consider a limit case where the maximum number of stages grows very large and discuss changes in beliefs and efforts across the stages on an intuitive basis. If the opponents’ effort choices remain informative about the type distribution, the impact of the own type on the players’ beliefs in stage s becomes less and less important in later stages, as the number of signals obtained increases rapidly (the opponent’s effort in stage s is not only informative about this opponent’s valuation but also about the opponent’s experience in previous stages). In the limit case after a sufficient number of stages, the heterogeneity in beliefs should disappear and all players’ beliefs about the share of players with a high prize valuation should converge to the true share ${\pi }_{\omega }$ (where $\omega \in \{\overline{\omega },\underline{\omega }\}$ ).Footnote ¹⁷ Moreover, the players anticipate that their opponent will have the same beliefs with probability (close to) one. As a consequence, the correlation between a player’s own effort and the average effort she expects from her stage s opponent identified in (12) and (13) above vanishes in later stages:

(14) $\begin{array}{c}\underset{s\to \infty }{lim}{E}_{i,s}\left(\left(x_{-i,s}\right),v_i,=,v_H\right)=\underset{s\to \infty }{lim}{E}_{i,s}\left(\left(x_{-i,s}\right),v_i,=,v_L\right).\end{array}$

Similarly, the average equilibrium effort of high-valuation (low-valuation) players converges to the equilibrium effort of high-valuation (low-valuation) players in a contest in which the players have common beliefs about the type distribution.

Proposition 5

Suppose that exit is possible at the end of stage 1. For an exit payment $b\in [b_L,b_H]$ , there is a perfect Bayesian equilibrium in which all players with valuation $v_L$ exit and all players with valuation $v_H$ do not exit, where $b_L$ and $b_H$ are given by

(15) $\begin{array}{c}{b}_L\equiv \left(q,\frac{max\{{\tilde{x}}_L,\underline{x}\}}{max\{{\tilde{x}}_L,\underline{x}\}+\frac{q{v}_H}{4}},v_L,-,max,\{{\tilde{x}}_L,\underline{x}\}\right)\sum _{t=2}^n{\left(1,-,q\right)}^{t-2}\end{array}$

with ${\tilde{x}}_L\equiv q\left(-,v_H,+,2,\sqrt{v_L{v}_H}\right)/4$ and

(16) $\begin{array}{c}{b}_H\equiv \frac{q{v}_H}{4}\sum _{t=2}^n{\left(1,-,q\right)}^{t-2}>b_L.\end{array}$

At stages $s=2,\dots ,n$ (if reached), all players believe that their opponent has a valuation $v_H$ with probability one and equilibrium efforts are equal to

(17) $\begin{array}{c}{x}_s^{\ast }=\frac{q{v}_H}{4}.\end{array}$

2.4 Summary of the main predictions

The theoretical analysis provides the basis of four main testable predictions. First, ignoring potential type heterogeneity and population uncertainty, efforts should be constant across all stages and beliefs should be type-independent (Proposition Footnote ¹). Second, if there is unobservable heterogeneity in the (intrinsic) motivation to win and players are ex ante uncertain about the distribution of these player types in the population, then the individual beliefs about the opponent’s effort are positively correlated with the own effort in early stages of the game. The correlation should become weaker in later stages of the game (compare Lemma Footnote ¹ and Proposition 6 in Section B.1 of the Online Appendix as well as the discussion around (14)). Third, if the true type distribution consists of more weak types than expected (with a low intrinsic motivation), weak types’ effort should go up and strong types’ effort should go down in later stages, as compared to stage 1 (Proposition Footnote ⁴). The opposite dynamics should prevail if the population consists of more strong types than expected. Forth, if exit is possible at the end of stage 1, weak types should exit and strong types should remain so that average effort in stages $2,\dots ,n$ is strictly higher than average stage 1 effort (Proposition Footnote ⁵).

The intuition for the dynamics of efforts closely follows a standard contest logic which is due to the non-monotonicity of best-reply functions. Weak types should be discouraged when learning that there are many strong opponents, and encouraged when learning that there are many weak opponents. Strong types should become more competitive when learning that there are many strong opponents, and should be “appeased” when learning there are many weak opponents. Together with the direction in which the beliefs are adjusted in the respective population, this explains the basic mechanism behind Proposition Footnote ⁴.

3.1 Treatments

The baseline experimental treatment BASE corresponds to the theory framework outlined above and investigates the importance of accounting for unobserved heterogeneity in preference types and belief formation, as opposed to the benchmark prediction based on complete information. In the experiment, the individuals compete in up to $n=5$ stages about a prize of monetary value $v=450$ by choosing investments $x_{i,s}$ from the set $\left(1,,,2,,,\dots ,,,450\right)$ at each stage s that is reached. Together with the choice of the own effort, each individual has to state the effort she expects from her opponent at this stage (as a number between 1 and 450); the stated beliefs $E_{i,s}\left(x_{-i,s}\right)$ are not displayed to other players. After the effort choices have been made at stage s, the individuals observe the investment $x_{-i,s}$ of their opponent and a “lottery wheel” determines whether one of the two players is allocated the prize or whether the game proceeds to the next stage; this outcome is observed, too.Footnote ²¹ The exogenous probability that the prize is allocated in a given stage is $q=1/3$ , which is supposed to balance a reasonable chance of winning the prize with a sufficiently high probability of continuation and, hence, possible dynamics. At each stage, the individuals are randomly re-matched in pairs. Once the game ends because the prize has been allocated or stage $s=5$ has been completed without prize allocation, each individual is displayed her own payoff.

This design with random matching as well as anonymity and non-identifiability of participants stays close to the theory as long as the subjects in the laboratory do not believe that their actions have informational content that feeds back into their own future encounters. The probability that, in a given session, a player interacted with the same player more than once is not zero in our setup, but the respective player would not know if/when meeting a particular opponent again, which should make quasi-repeated play effects rather unlikely.

Whereas dynamic conflict games typically involve explicit or implicit participation decisions, the BASE treatment removes such considerations by design. As the main experimental variation, the EXIT treatment therefore adds the possibility of exit and, hence, an explicit continuation decision. Based on this experimental variation, we can identify possible effort escalation caused by self-selection as a consequence of unobserved preference heterogeneity. As in the modified theory framework described above, the individuals have the option to exit the game at the end of stage 1, after observing the stage 1 efforts and outcome and in case the prize has not yet been allocated at stage 1.Footnote ²² Individuals make this choice between “exit” and “remain” simultaneously and independently. Denote the stage 1 pair of players by $\left(i,,,-,i\right)$ . If both i and $-i$ choose to exit then the game ends for both individuals with an exit payment of 60 points each (minus the individual cost of stage 1 effort). If both individuals i and $-i$ choose to remain then both enter into stage 2 (where new pairs of subjects are randomly formed). If one individual chooses to exit and her stage 1 opponent chooses to remain then a coin flip decides on whether both subjects exit or enter into stage 2.Footnote ²³ Apart from adding this exit option, the sequence of actions in the EXIT treatment is exactly as in the BASE treatment. The payment in case of exit is chosen to be lower than the equilibrium expected continuation payoff in the benchmark case with symmetric players who maximize their monetary payoffs so that in the latter case no player should exit in equilibrium.

Our choice of symmetric monetary incentives allows to attribute any heterogeneity in behavior to differences in unobserved (preference) characteristics. This comes at the cost of not being able to identify different types of players based on an observable objective function but having to rely on individual choices in order to distinguish different behavioral types of players. This latter approach to classifying types, which is common when one expects factors beyond monetary payoffs to matter, would be appropriate even when imposing heterogeneity in extrinsic (incentivized) motivations as another layer of heterogeneity: even with imposed differences in effort costs, for instance, the sets of individuals with identical monetary incentives would have to be expected to differ along important preference characteristics. The vast majority of contest experiments has shown that behavior cannot be understood without incorporating motives beyond monetary payoff maximization.Footnote ²⁴

3.2 Experimental procedures

The experiment was conducted at econlab Munich in two waves and 19 sessions in total (with typically 24 subjects per session). A first wave in April 2016 involved 4 sessions of each treatment BASE and EXIT; in this wave, the elicitation of beliefs was not incentivized to reduce complexity. A second wave in May 2019 involved 4 sessions of the BASE treatment for which the elicitation of beliefs was incentivized, plus 4 BASE sessions with nonincentivized beliefs and 3 EXIT sessions to ensure comparability of the two waves. The subjects (422 in total) were typically students of Munich universities.Footnote ²⁵ Each subject took part in exactly one session. In each treatment, the respective multi-stage contest was played for 15 times. In other words, each subject played the same game (with up to 5 stages s) in 15 rounds r.

At the beginning of each session each subject was shown a video on the computer screen in which the experimental instructions (also distributed as hard copy) were read aloud. Then, the subjects had to answer a few control questions to ensure they understood the rules of the experiment. After the 15 rounds of the main experiment, we conducted an extended post-experimental questionnaire which, apart from socioeconomic information and questions about the experiment, elicited measures for risk preferences, distributional preferences, ambiguity aversion, loss aversion, and cognitive reflection. At the end of the experiment, 3 out of the 15 rounds were randomly selected for payment and a subject’s total points won minus her investments $x_{i,s}$ were summed up. In the sessions in which the stated beliefs were incentivized, one further round was selected (different from the 3 rounds for which the contest outcome was paid) and a subject obtained 450 points if her stated beliefs in this additional round deviated, on average, by 5 points or less from the actual opponent effort (that is, if on average $\left(E_{\mathrm{irs}},\left(x_{-irs}\right),-,x_{-irs}\right)\le 5$ in this round). Moreover, one of the incentivized post-experimental tasks were randomly selected for payment. The resulting amount of money was converted to Euros at the rate 50 : 1 and added to (or subtracted from) an endowment of 10 Euros. On average a session lasted 90 minutes and the average payment was 17.70 Euros plus a show-up fee of 6 Euros.

For the experimental sessions, the randomization of whether or not in a given stage $s\in \left(1,,,\dots ,,,5\right)$ of round $r\in \left(1,,,\dots ,,,15\right)$ the prize would be allocated (with the exogenous probability $q=1/3$ ) was conducted (but not announced) before the start of the first session and was kept the same across all treatments, sessions, and subject pairs.Footnote ²⁶ In other words, the number of stages to be played within round $r\in \left(1,,,\dots ,,,15\right)$ was the same for all subjects. This ensures that learning about the game and the numbers of signals obtained about the distribution of types are identical across treatments and sessions at any stage s of round r. The random re-matching took place in subgroups (typically 8 subjects, although the precise size of the matching groups was not made explicit) in order to gain more independence of observations and allow us to investigate learning dynamics across different populations.Footnote ²⁷