Francesco Feri (Innsbruck) MA Mel é ndez (M á laga) Giovanni Ponti (UA-UniFE) Fernando Vega (IUE)

1. Error Cascades in Positional LearningAn Experiment on the Chinos Game Francesco Feri (Innsbruck) MA Meléndez (Málaga) Giovanni Ponti (UA-UniFE) Fernando Vega (IUE) 2007ESA - LuissRM- 30/6/07

2. Perfectlyobserved Motivation • Situations where agents have to take public decisions in sequence, along which • Actions • Identities • Private valuable information is (may be) revealed through actions • Financial markets • Technological adoptions • Firms’ business strategies (uncertain market conditions) • Observational (“Positional”) Learning

3. Related literature

4. Feri et al. (2006): the “Chinos’ Game” • Each player hides in her hands a # of coins • In a pre-specified order players guess on the total # of coins in the hands of all the players • Information of a player Her own # of coins Predecessors’ guesses + • Our setup → simplified version: • 3 players • # of coins in the hands of a player: either 0 or 1 • Outcome of an exogenous iid random mechanism (p[s1=1]=.75) • Formally: multistage game with incomplete information

5. The “Chinos’ Game”: Game-Form (2-players)

6. Outcome function • All players who guess correctly win a prize: • All Win Game (AWG) • Players’ incentives do not conflict • Unique Perfect Bayesian Equilibrium: Revelation • Perfect signal of the private information • After observing each player’s guess, any subsequent player can infer exactly the number of coins in the predecessors’ hands.

7. WPBE for the Chinos Game • Players:i  N  {1, 2, 3} • Signal (coins): si S  {0, 1} • Random mechanism: P(si = 1) = ¾ (i.i.d.) • Guesses: gi G  {0, 1, 2, 3} • Information sets: I1  S I1=s1 I2  S x G I2=(s2, g1) I3 S x G2 I3=(s3, g1, g2) • PBE: revelation • g1 = s1 + 2 • g2 = g1 + s2 - 1 • g3 = g2 + s3 - 1

8. “Reasonable” beliefs • (Out-of-equilibrium) beliefs are as such that later movers always belief that out-of equilibrium guesses are associated with the signal that “would have yielded” the highest expected payoff

9. Experimental design • Sessions: 4 held in May 2005 • Subjects: 48 students (UA), 12 per session (1 1/2 hour approx., € 19 average earning) • Software: z-Tree (Fischbacher, 2007) • Matching: Fixed group, fixed player positions • Independent observations: 4x(12/3=4)=16 • Information ex ante: private signal • Information ex post: everything about about everything (signals & choices) about group members • Random events: everything (i.e. signals) iid.

10. Descriptive results: Outcomes • Frequency of right guesses increases with player position • Difference between theoretical and actual frequences also increases with player position

11. Descriptive results: Behavior (player 1) • Behavioral strategies follow expected payoffs • Better play when s1=0 (???)

12. Descriptive results: Behavior (Player 2) • Adherence with equilibrium much higher when g1=3

13. Descriptive results: Behavior (Player 3) • Adherence with equilibrium much higher when g1=3

14. Towards a theory of “error cascades” •  is a measure how subjects do well from their own perspective •  is a measure how subjects do well from their followers’ perspective • This interpretation (may) fall shortout of the equilibrium path

15. Towards a theory of “error cascades”

16. Notional learning Towards a theory of “error cascades” “… Any other view risk relegating rational players to the role of the “unlucky” bridge expert who usually loses 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 learning notionally if they play a best-response to the equilibrium strategy of their opponent

17. Optimal learning Towards a theory of “error cascades” • Players are learning optimally if they play a best response to their predecessors’ strategies (that they can infer by past experience)

18. Thetas & betas: Player 2

19. Thetas & betas: Player 3

20. Error cascades in the Chinos Game

21. Error cascades in the Chinos Game

22. Error cascades in the Chinos Game

23. (A)QRE: A Theory of Error Cascades • The basic question: why error cascades? • Assume that subjects' choices are also affected by other (unmodeled) external factors that make this process intrinsically noisy • Why? Complexity of the game, limitation of subjects' computational ability, random preference shocks, etc… • A “classic” model of (endogenous) noise: McKelvey and Palfrey’s [1995] Quantal Response Equilibrium • The QRE approach is applied to the “Agent Normal Form” (McKelvey & Palfrey, EE 1998)

24. It is essentially a QRE IN BEHAVIORAL STRATEGIES (Logit) Quantal Response Equilibrium (QRE) • In a (A)QRE, (full support) behavioral strategies follow expected payoffs:

25. Estimating individual QRE noise parameters (I) • Individual (static) estimates • Common beliefs assumed • All (24) observations considered

26. Player 1’s QRE

27. Player 2’s QRE

28. Prop. 4.1 Prop. 4.2 Prop. 5 Player 2’s QRE

29. 2(3) 2(2) 2 1 Error cascades along the equibrium path (g1=2 & s2=1)

30. 2(3) 2(2) 2 1 Error cascades along the equibrium path (g1=3 & s2=1)

31. Error cascades on the equibrium path: Player 2 (s2=1)

32. Error (QRE) cascades: Player 3

33. Further Research: Conflicting interest • Constant sum games • One and only one player in the group wins the prize • Agents’ incentives → Pure conflict • First win game (FWG) • Winner → the player who first guesses correctly • If no one guess right → the prize goes to player 3 • Equilibrium → revelation (but no repetition constraint) • Last win game (LWG) • Winner → the last player who guesses correctly • If no one guess right → the prize goes to player 1 • Equilibrium → uninformative pooling • Last, but not least (…) • Positional learning with noise (Carbone and Ponti, 2007)