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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:
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1x
Introductory examples (Goeree and Holt)
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
Introductory examples (Goeree and Holt)
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
Introductory examples (Goeree and Holt)
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Introductory examples (Goeree and Holt)
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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
Introductory examples (Goeree and Holt)
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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
Introductory examples (Goeree and Holt)
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,
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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
1x
Structural approach 1
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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
Structural approach 1
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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
Structural approach 1
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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
Structural approach 2
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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
Alternative explanations: equilibrium selection
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
Alternative explanations: equilibrium selection
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
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 .
;, 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
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)
ix c
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Behavioral adjustment rules 1Behavioral adjustment rules 1
Behavioral foundations
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
Behavioral foundations
(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
Games with strategic complementarities
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
Games with strategic complementarities
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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