a unified approach to comparative statics puzzles in experiments armin schmutzler

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A unified approach to comparative statics puzzles in experiments

Armin SchmutzlerUniversity of Zurich, CEPR, ENCORE

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Introduction 1Introduction 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)

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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 2Introduction 2

Starting point: „Ten little treasures of game theory and ten intuitive contradictions“ (Goeree and Holt 2001)

Introduction

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Kreps game:Kreps game:

50

Equilibria:

01 x 200, 50 0, 45 10, 30 20, -250

0, -250 10, -100 30, 30 , 405/6

02 x 12 x 22 x 32 x

11 x

0,0,03.0,97.0,92.0,08.0;3,1;0,0

Introductory examples (Goeree and Holt)

0.32 0

0.96 0.84

0

11p 32p

300

Observation:

5

1x


Player 1 x1 = 0 x1 = 1 Player 2

(80,50) x2 = 0 x2 = 1

(90,70) (20,10 + θ)

A common-interest proposal game

Unique SPE

0.16 0

0.52 0.25

60,0021 xx

0

for

Observation

11p 12p

58

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Strategy spaces:

Payoffs:

Standard theory:unique equilibriumsurvives iterated elimination of dominated strategies

Traveler‘s dilemma (Basu 1994)Traveler‘s dilemma (Basu 1994)

jijijii xxsignxxxx ,min;,

300,...,18021 XX

1; R

18021 xx


Observations:Actions are higher for lower fines (high )

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1x

2x ;21 xR

184

183

182

181

180

180 181 182 183

;12 xR

o45


8


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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 examplesSubjective summary of examples


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Assumptions:two-player games, parameterized byPayoff functionparameter space partially orderedstrategy space is

independent of parameter compact

Notation:

General set-up and notationGeneral set-up and notation

;, jii xx

;,;,;,, 2112112111 xxxxxxx LHLH

Notation

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Kreps game:Kreps game:

50

Equilibria:

01 x 200, 50 0, 45 10, 30 20, -250

0, -250 10, -100 30, 30 , 405/6

02 x 12 x 22 x 32 x

11 x

0,0,03.0,97.0,92.0,08.0;3,1;0,0


0.32 0

0.96 0.84

0

11p 32p

300

Observation:

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Observation: non-decreasing in (ID)

non-decreasing in (SUP)

An intuitive explanation for the Kreps gameAn intuitive explanation for the Kreps game

;;, jLi

Hii xxx

01 x

Incremental Payoffs

-200, -5 10, -15 20, -280

-, 150 -, 130 -, -

,30

02 x 12 x 22 x 32 x

11 x

;;, j

Li

Hii xxx

jx

Thusnon-negative direct effect of on (reaction function shifts out)

these effects are mutually reinforcing (non-decreasing reaction function)

21, xx

Structural approach 1

105/6,

13

1x

2x

0

1

2

3

4

5

1 2 3 4 5

LxR ;21

HxR ;21

HxR ;12

LxR ;12

1x


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A more formal explanationA 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


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Other supermodular games in GHOther 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


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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: SummaryStructural approach 2: Summary


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Many of the above examples have alternative explanations: equilibrium selection theories quantal-response equilibrium

Goal: Explore the relation to my approach

Alternative Explanations: OverviewAlternative 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)

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Effort coordination: exampleEffort coordination: example

0,0 0,-c

-c,0 1-c,1-c

02 x 12 x

11 x

01 x

Alternative explanations: equilibrium selection

Structural approach:

(ID) and (SUP) hold; Thus non-decreasing in ix c

Standard approach:

PSE constant, MSE decreasing in c!

contradicts evidence

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Risk dominance in symmetric 2x2-gamesRisk dominance in symmetric 2x2-games


Suppose

Equilibrium set for

1,0iX

1,1,0,0 LHLH ;,

Proposition: If (ID) holds and risk dominance selects (1,1) for , it also does so for .L H

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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 maximizationRelation to potential maximization

i

i

i xx

V


HH xx ,

HL xx

LH LL xx ,

LxxV ;, 21

Potential function: V such that

E

HxxV ;, 21

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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 standWhere we stand

Behavioral foundations

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Assume

where

satisfies (SUP) and (ID)

satisfies (SUP) and (ID)

Nash equilibria of perturbed gamesNash equilibria of perturbed games

;jii xxg

;,;,;,ˆ jiijiijii xxgxxxx


Then the game with modified objective functions still satisfies (ID) and (SUP).

Therefore: For the perturbed game, the equilibrium is non-decreasing in .

;, jii xx

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Effort coordination: Modified exampleEffort coordination: Modified example

0,0 k,-c

-c,k 1-c,1-c

02 x 12 x

11 x

01 x


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)

ix c

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Behavioral adjustment rules 1Behavioral adjustment rules 1


Idea: Comparative statics does not require reference to any equilibrium concept

Alternative:

model of adjustment to changeadjustment as dynamic processperiod 1 captures direct effectremaining periods capture indirect effects

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Assumption (ADJ): such that:

Behavioral adjustment rules 2Behavioral adjustment rules 2

1t

ti

LH axx ii


(ADJ1) Suppose for :

Then

(ADJ2) Suppose is supermodular in .

Then implies .

LH .;,,;,, L

iiiiH

iiii xxxxxx

01 ia

;, jii xx

0tia 01 t

ia

Proposition: If (SUP), (ID) and (ADJ) hold, the adjustment process converges to such that

ji xx ,

Li

Hi xx 21, xx

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SummarySummary

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

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LimitationsLimitations

no point predictions

not applicable in some cases

will probably fail in some cleverly designed experiments

Conclusions

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Strategy Spaces:

Payoffs:

Theory:unique equilibriumsurvives iterated elimination of dominated strategies

Traveler‘s dilemma (Basu 1994)Traveler‘s dilemma (Basu 1994)

jijijii xxsignxxxx ,min;,

300,...,18021 XX

1; R

18021 xx

Games with Strategic Complementarities

Observations:Actions are higher for lower fines (high )

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1x

2x ;21 xR

184

183

182

181

180

180 181 182 183

;12 xR

o45

Games with strategic complementarities

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Violation of SupermodularityViolation of Supermodularity

202

203

0 1

201jx 202jx 203jx 204jx

;,202 ji x


1

;,203 ji x

;,202,203 ji x

201 202

203

202

202201

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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

ExplanationExplanation

;ix

iR

;, 21 xxi

21 , xx

21 , xx

1R 2R


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1x

2x

0

1

2

3

4

5

1 2 3 4 5

LxR ;21

HxR ;21

HxR ;12

LxR ;12

Games with strategie complementarities

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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 complementaritiesGH puzzles and strategic complementarities

Games with strategie complementarities

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Set-Up (GH 01, Ochs 95):

Equilibrium:

Observation

Unilateral shifts of reaction functions: matching penniesUnilateral shifts of reaction functions: matching pennies

,40;1,021 xx

40

1,40

2

1,

2

1 *2

*1

40, 80

40, 80 80, 40

02 x 12 x

11 x

01 x 40,

Other Games

player 1‘s action decreasing in

player 2‘s action increasing in

:320,80,44

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1

0.875

0.5

0.091

0.5 1

21

1 320;21 xR

80;21 xR

44;21 xR

;12 xR

11

Other Games

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LH xRxR ;; 1212

LxR ;21

HxR ;212x

Hx2Lx2

Lx1Hx1 1x

Other Games

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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.

ExplanationExplanation

21, xx ;, 211 xx

;1x

;, 212 xx

21, xx

;, 211 xx

;, 212 xx

1x 2x

Other Games

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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 equilibriumQuantal response equilibrium

Alternative explanations: quantal response equilibrium

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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 equilibriumStructural approach vs. quantal response equilibrium

Alternative explanations: quantal response equilibrium

a unified approach to comparative statics puzzles in experiments armin schmutzler

Documents

higher nash equilibrium

gh examples

notationkreps game

introductionkreps game

unified approach

id nondecreasing

nash equilibriumobservation

common explanation