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A unified approach to comparative statics puzzles in experiments Armin Schmutzler University of Zurich, CEPR, ENCORE. Introduction 1. Introduction. Issue: Can we learn anything from game-theoretic reasoning based on Nash equilibrium even when literal application of concept fails?. Here:

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slide1

A unified approach to comparative statics puzzles in experiments

Armin Schmutzler

University of Zurich, CEPR, ENCORE

introduction 1
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)
introduction 2
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
kreps game
Kreps game:

Introductory examples (Goeree and Holt)

Equilibria:

Observation:

slide5

Introductory examples (Goeree and Holt)

A common-interest proposal game

Unique SPE

for

Observation

traveler s dilemma basu 1994
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 )
subjective summary of examples
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
general set up and notation
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:

kreps game1
Kreps game:

Introductory examples (Goeree and Holt)

Equilibria:

Observation:

an intuitive explanation for the kreps game
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)
a more formal explanation
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
other supermodular games in gh
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

structural approach 2 summary
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.
alternative explanations overview
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)
effort coordination example
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
risk dominance in symmetric 2x2 games
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 .

relation to potential maximization
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 .

where we stand
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
nash equilibria of perturbed games
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 .

effort coordination modified example
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)
behavioral adjustment rules 1
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
behavioral adjustment rules 2
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

summary
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
limitations
Limitations

Conclusions

  • no point predictions
  • not applicable in some cases
  • will probably fail in some cleverly designed experiments
traveler s dilemma basu 19941
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 )
violation of supermodularity
Violation of Supermodularity

Games with strategic complementarities

explanation
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

gh puzzles and strategic complementarities
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
unilateral shifts of reaction functions matching pennies
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
explanation1
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.

quantal response equilibrium
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)
structural approach vs quantal response equilibrium
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