A unified approach to comparative statics puzzles in experiments Armin Schmutzler

1. A unified approach to comparative statics puzzles in experiments Armin Schmutzler University of Zurich, CEPR, ENCORE

2. Introduction 1 Introduction Issue: Can we learn anything from game-theoretic reasoning based on Nash equilibrium even when literal application of concept fails? • Here: • consider experiments where • Nash point predictions do not hold • parameter changes affect behavior even though Nash equilibrium suggests no change • show that suitable modification of standard theory can predict observed treatment effects (without giving point predictions)

3. Introduction 2 Introduction Starting point: „Ten little treasures of game theory and ten intuitive contradictions“ (Goeree and Holt 2001) • Set-Up: • ten pairs of experiments that differ in parameter • Theory: • does not change Nash equilibrium • Observation: • shift of affects behavior • Contribution: • provide unified explanation for seven of these puzzles

4. Kreps game: Introductory examples (Goeree and Holt) Equilibria: Observation:

5. Introductory examples (Goeree and Holt) A common-interest proposal game Unique SPE for Observation

6. Traveler‘s dilemma (Basu 1994) Introductory examples (Goeree and Holt) • Strategy spaces: • Payoffs: • Standard theory: • unique equilibrium • survives iterated elimination of dominated strategies • Observations: • Actions are higher for lower fines (high )

7. Introductory examples (Goeree and Holt)

8. Introductory examples (Goeree and Holt)

9. Subjective summary of examples Introductory examples (Goeree and Holt) • In all three cases, • has no effect on equilibrium set • observed actions increase with • Task: • Find a common explanation of observed comparative statics Note: • In Kreps game, this is closely related to selection issue • Other people have provided other explanations

10. General set-up and notation Notation • Assumptions: • two-player games, parameterized by • Payoff function • parameter space partially ordered • strategy space is • independent of parameter • compact Notation:

11. Kreps game: Introductory examples (Goeree and Holt) Equilibria: Observation:

12. An intuitive explanation for the Kreps game Structural approach 1 Incremental Payoffs • Observation: • non-decreasing in (ID) • non-decreasing in (SUP) • Thus • non-negative direct effect of on (reaction function shifts out) • these effects are mutually reinforcing (non-decreasing reaction function)

13. Structural approach 1

14. A more formal explanation Structural approach 1 • Proposition: (Milgrom and Roberts 1990) • Suppose (SUP) and (ID) hold. Then: • A smallest and largest pure strategy equilibrium exist • Both are non-decreasing functions of • Summary of Kreps game: • Subjects choose higher actions for higher • Nash equilibrium in Kreps game is independent of • Under (SUP) and (ID), Nash equilibrium is non-decreasing in

15. Other supermodular games in GH Structural approach 1 Three other GH examples can be explained like the Kreps game, namely • The extended coordination game • The common-interest proposal game • The conflicting-interest proposal game Issue now: Extend this approach to other games with strategic complementarities

16. Structural approach 2: Summary Structural approach 2 • Main point: • Comparative statics results such as Proposition 1 hold for instance in • games with strategic complementarities • games where strategic interactions differ across players and parameter affects only one payoff • Implication: • Three other GH-examples are consistent with the structural approach.

17. Alternative Explanations: Overview Alternative explanations • Many of the above examples have alternative explanations: • equilibrium selection theories • quantal-response equilibrium • Goal: Explore the relation to my approach • structural approach is closely related to risk-dominance and potential maximization • can sometimes revert implausible predictions of standard approach • Examples: • Effort coordination games (Anderson et al. 2001) • Other 2 x 2-coordination games(Guyer and Rapoport 1972, Huettel and Lockhead 2000, Schmidt et al. 2003)

18. Effort coordination: example Alternative explanations: equilibrium selection • Standard approach: • PSE constant, MSE decreasing in c! • contradicts evidence • Structural approach: • (ID) and (SUP) hold; Thus non-decreasing in

19. Risk dominance in symmetric 2x2-games Alternative explanations: equilibrium selection Suppose • Equilibrium set for Proposition: If (ID) holds and risk dominance selects (1,1) for , it also does so for .

20. Relation to potential maximization Alternative explanations: equilibrium selection Potential function: V such that Proposition: Consider a symmetric game satisfying (ID). Suppose the set of PSE is identical for parameters . If maximizes the potential function on E, and maximizes , then .

21. Where we stand Behavioral foundations • So far: • Structural approach often provides predictions that are consistent with experimental evidence • But why? • Possible explanations: • Actual payoffs are perturbations of monetary payoffs that leave comparative-statics unaffected • Players react to parameter changes using plausible adjustment rules

22. Nash equilibria of perturbed games Behavioral foundations • Assume • where • satisfies (SUP) and (ID) • satisfies (SUP) and (ID) Then the game with modified objective functions still satisfies (ID) and (SUP). Therefore: For the perturbed game, the equilibrium is non-decreasing in .

23. Effort coordination: Modified example Behavioral foundations • k>0 (anti-social preferences): • Game still satisfies (ID) and (SUP) • Thus non-decreasing in • c<1-k: multiple equilibria; c>1-k: only (0,0)

24. Behavioral adjustment rules 1 Behavioral foundations Idea: Comparative statics does not require reference to any equilibrium concept • Alternative: • model of adjustment to change • adjustment as dynamic process • period 1 captures direct effect • remaining periods capture indirect effects

25. Behavioral adjustment rules 2 Behavioral foundations Assumption (ADJ): such that: (ADJ1) Suppose for : Then (ADJ2) Suppose is supermodular in . Then implies . Proposition: If (SUP), (ID) and (ADJ) hold, the adjustment process converges to such that

26. Summary Conclusions • Paper resolves some contradictions between „standard game theory“ and the lab • Proposes a way to derive directions of change when mechanical application of Nash concept suggests no change (Structural Approach) • Applicable to comparative statics and multiplicity problems

27. Limitations Conclusions • no point predictions • not applicable in some cases • will probably fail in some cleverly designed experiments

28. Traveler‘s dilemma (Basu 1994) Games with Strategic Complementarities • Strategy Spaces: • Payoffs: • Theory: • unique equilibrium • survives iterated elimination of dominated strategies • Observations: • Actions are higher for lower fines (high )

29. Games with strategic complementarities

30. Violation of Supermodularity Games with strategic complementarities

31. Explanation Games with strategic complementarities Traveler‘s dilemma has the following properties: (B1) well-defined reaction functions (B2) non-decreasing reaction functions (B3) has increasing differences in (B4) For each , unique equilibrium (B5) lies above (only) to the right of the equilibrium For any such game, is weakly increasing in

32. Games with strategie complementarities

33. GH puzzles and strategic complementarities Games with strategie complementarities • so far: five of the GH puzzles solved • GSC-argument carries over to an auction game • argue next: Embedding Principle can be applied to another example that is not GSC

34. Unilateral shifts of reaction functions: matching pennies OtherGames • Set-Up (GH 01, Ochs 95): • Equilibrium: • Observation • player 1‘s action decreasing in • player 2‘s action increasing in

35. OtherGames

36. OtherGames

37. Explanation OtherGames Matching pennies has the following properties: (C1) well-defined reaction functions (C2) is supermodular in (C3) is supermodular in (C4) satisfies increasing differences in (C5) is independent of For each such game, is weakly decreasing, is weakly increasing.

38. Quantal response equilibrium Alternative explanations: quantal response equilibrium • Definition: • In a quantal response equilibrium players best-respond up to a stochastic error • Belief probabilities used to determine expected payoffs match own choice probabilities • Applications: • Traveler‘s dilemma (Anderson et al. 2001, Capra et al. 1999) • Effort coordination games (Anderson et al. 2001)

39. Structural approach vs. quantal response equilibrium Alternative explanations: quantal response equilibrium • Comparison: • Quantal response comparative statics also exploits structural properties, e.g., • local payoff property of expected payoff derivative • (ID)-like property based on expected payoffs • Advantage of structural approach : • (ID) and (SC) observable from fundamentals • no symmetry assumption • no local payoff property required