1. This paper reports an experimental study based on the popular Chinos game, in which three players, arranged in sequence, have to guess the total number of coins lying in their hands to win a fixed prize. Since guesses are public, players should be able to decode, through their predecessors’ guesses, their private information. In this respect our evidence shows that, despite the remarkable frequency of equilibrium outcomes, deviations from optimal play are also significant, specially in some groups. In this case we find that the probability of a mistake, for any given player position, is increasing in the probability of a mistake of her predecessors. Hence, this fact results in what we call error cascades, i.e. errors accumulate through the guessing sequence. Imperfect Learning and Error Cascades in Sequential Guessing Games: An Experiment Francesco Feri (Univ. Venezia) Giovanni Ponti (Univ. Alicante and Univ. Ferrara) Miguel A. Meléndez-Jiménez (Univ. Málaga) Fernando Vega-Redondo (Univ. Alicante and Univ. Essex) Francesco Feri (Univ. Venezia) Giovanni Ponti (Univ. Alicante and Univ. Ferrara) Miguel A. Meléndez-Jiménez (Univ. Málaga) Fernando Vega-Redondo (Univ. Alicante and Univ. Essex) ABSTRACT 1.- Introduction 5.- Learning Analysis Motivation  Situations where agents have to take public decision sequentially and under uncertainty: o Actions and identities are perfectly observed.  Each agent may own private valuable information. Economic examples: o Financial markets o Technological adoptions o Firms business’ strategies under uncertain market conditions. We have fixed groups where agents repeatedly play with the same opponents. Therefore, subjects should be able to learn their predecessors’ patterns of behavior (through past experience), even if the preceding players consistently deviate from the equilibrium. We investigate whether agents are able to react optimally to the behavior they observe from their predecessors in the group. “… Any other view risk relegating rational players to the role of the “unlucky” bridge expert who usually looses but explains that his play is “correct” and would have led to his winning if only the opponents had played correctly …”Binmore (1987) Players are optimal when they play a best response with respect to their predecessor’s strategies (that they infer throughout the past experience) → PERFECT LEARNING 6.- Conclusion Objectives  To analyze agents’ behavior in signaling games with no conflict of incentives among agents.  To study agents observational learning when the game is repeated. 2.- The Model The Chinos game  Three players: n = 3.  Each player hides in her hands a # of coins: s i  {0, 1} i=1,2,3. (i.i.d. draws from an exogenous random mechanism)  In a pre-specified order players guess on the total # of coins in the hands of all the players: g i  {0, 1, 2, 3}. Information of a player Her own # of coins + Predecessors’ guesses  Payoffs: All players who gets correctly win a prize (no conflict of incentives).  Multistage game with incomplete information. Equilibrium  Unique equilibrium: revelation.  A player’s equilibrium guess is a perfect signal of her private information.  After observing each player’s guess, any subsequent player can infer exactly the number of coins lying in the predecessor’s hand. 3.- Experimental Design  48 students from the University of Alicante (2002).  4 sessions (12 subjects per session).  Software: z-tree (Fischbacher, 1999)  In each session: 4 fixed groups of three players (to study learning).  Each subject played anonymously 20 rounds of the game with the same opponents.  Player position fixed.  After each round subjects were informed about all relevant payoff variables in their groups  History table: subjects could keep track of the history of their groups.  Payoffs: 1000 ptas. to show up. The fixed prize for each round was 50 ptas. (166ptas=1Euro).  Random device: Prob(s i =1) = ¾ (i.i.d.) 4.- Descriptive Results Player Frequency of guessing right (%) 140.5 (56) 250.3 (75) 361.1 (100) Outcomes  Aggregating for player position: frequency of getting the prize  In brackets: the theoretical predictions.  Frequencies increase w.r.t. player position.  But also the difference between observed frequencies and theoretical ones increases w.r.t. player position. Player Frequency of equilibrium play (%) 159.2 265.8 363.2 Behavior  Aggregating for player position: frequency of equilibrium play.  The experimental data qualitatively reproduces the theoretical prediction (high % of eq. play).  Nevertheless, deviations from the equilibrium are also significant.  We rationalize deviations by a learning analysis. Econometric Analysis  From player 2’s perspective: 1)Frequentist estimate (  1 ) a la fictitious play of player 1’s behavioral strategy 2)For each player 1’s guess, this induces: b 1 = Prob [player 1 behaves optimally] 3)Define an indicator function for player 2: 1 if player 2 behaves optimally given the estimate  1 BR 2 = 0 otherwise  Analogously, for player 3, we define b 2 and BR 3 ---------------------------------------------- BR2 | Coef. Std. Err. z P>|z| -------------+-------------------------------- b1 | 2.747572.6980889 3.94 0.000 _cons | -1.239713.4699841 -2.64 0.008 Player 2’s analysis  (Panel data) logit regression (using rounds 11-20).  Prob(BR 2 =1) is increasing in b 1. Imperfect Learning Player 3’s analysis  (Panel data) logit regression (using rounds 11-20).  Prob(BR 3 =1) is increasing in b 2. -------------------------------------------- br3 | Coef. Std. Err. z P>|z| ------+--------------------------------------- b2 | 1.637885.5426834 3.02 0.003 _cons | -.3373284.3831164 -0.88 0.379 Imperfect Learning Imperfect Learning by player 2 Imperfect Learning by player 3 Result in ERROR CASCADES  We have designed an experiment to analyze behavior in situations where agents’ incentives do not conflict. Each agent has incentives to decode predecessor’s private information (and to take expected value for successors) in order to take the guess which maximizes her probability to win.  The equilibrium of the game qualitatively predicts subjects’ behavior in the experiment. Nevertheless, deviations are observed.  Since the game is repeated with fixed groups, we perform an econometric analysis to study the relationship between the probability to play a best response of a player and the probability with which her predecessor in the sequence performs optimally (given the past experience).  We find the presence of imperfect learning: the smaller the probability with which a player performs optimally, the lower the ability of her successor in the sequence to play a best response.  Imperfect learning of players result in error cascades: errors accumulate through the guessing sequence.  In a parallel work, Feri et al (2004), we experimentally analyze the situation where agents incentives conflict, by modifying the payoff scheme. Errors accumulate through the guessing sequence