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

2 Introduction 1 Issue: Can we learn anything from game-theoretic reasoning based on Nash equilibrium even when literal application of concept fails? Here: consider experiments where Introduction  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 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 Introduction 2 Starting point: „Ten little treasures of game theory and ten intuitive contradictions“ (Goeree and Holt 2001) Introduction

4 Kreps game: Equilibria: 200, 50 0, 4510, 30 20, , , , 30, Introductory examples (Goeree and Holt) Observation:

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

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

Introductory examples (Goeree and Holt)

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9 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 Subjective summary of examples Introductory examples (Goeree and Holt)

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

11 Kreps game: Equilibria: 200, 50 0, 4510, 30 20, , , , 30, Introductory examples (Goeree and Holt) Observation:

12 Observation: non-decreasing in (ID) non-decreasing in (SUP) An intuitive explanation for the Kreps game Incremental Payoffs -200, -5 10, -1520, , 150-, 130-, - Thus non-negative direct effect of on (reaction function shifts out) these effects are mutually reinforcing (non-decreasing reaction function) Structural approach 1

13 Structural approach 1

14 A more formal explanation Proposition: (Milgrom and Roberts 1990) Suppose (SUP) and (ID) hold. Then: i.A smallest and largest pure strategy equilibrium exist ii.Both are non-decreasing functions of Summary of Kreps game: 1. Subjects choose higher actions for higher 2. Nash equilibrium in Kreps game is independent of 3. Under (SUP) and (ID), Nash equilibrium is non-decreasing in Structural approach 1

15 Other supermodular games in GH 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 Structural approach 1

16 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. Structural approach 2: Summary Structural approach 2

17 Many of the above examples have alternative explanations:  equilibrium selection theories  quantal-response equilibrium Goal: Explore the relation to my approach Alternative Explanations: Overview Alternative explanations  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 0,00,-c -c,01-c,1-c Alternative explanations: equilibrium selection Structural approach: (ID) and (SUP) hold; Thus non-decreasing in Standard approach: PSE constant, MSE decreasing in c! contradicts evidence

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 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. Relation to potential maximization Alternative explanations: equilibrium selection Potential function: V such that

21 So far: Structural approach often provides predictions that are consistent with experimental evidence But why? Possible explanations: (1) Actual payoffs are perturbations of monetary payoffs that leave comparative-statics unaffected (2) Players react to parameter changes using plausible adjustment rules Where we stand Behavioral foundations

22 Assume where satisfies (SUP) and (ID) Nash equilibria of perturbed games Behavioral foundations 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 0,0k,-c -c,k1-c,1-c Behavioral foundations k>0 (anti-social preferences): Game still satisfies (ID) and (SUP) Thus non-decreasing in 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 Assumption (ADJ): such that: Behavioral adjustment rules 2 Behavioral foundations (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 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 Conclusions

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

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

Games with strategic complementarities

30 Violation of Supermodularity Games with strategic complementarities

31 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 Explanation Games with strategic complementarities

32 Games with strategie complementarities

33 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 GH puzzles and strategic complementarities Games with strategie complementarities

34 Set-Up (GH 01, Ochs 95): Equilibrium: Observation Unilateral shifts of reaction functions: matching pennies 40, 80 80, 40 Other Games player 1‘s action decreasing in player 2‘s action increasing in

Other Games

36 Other Games

37 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. Explanation Other Games

38 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) Quantal response equilibrium Alternative explanations: quantal response equilibrium

39 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 Structural approach vs. quantal response equilibrium Alternative explanations: quantal response equilibrium