(Non) Equilibrium Selection, Similarity Judgments and the

1
(Non) Equilibrium Selection, Similarity Judgments
and the Nothing to Gain / Nothing to Lose
Effect

• Jonathan W. Leland
• The National Science Foundation
• June 2007
• The research discussed here was funded by an
  Italian Ministry of Education Rientro dei
  Cervelli fellowship. Views expressed do not
  necessarily represent the views the National
  Science Foundation nor the United States
  government. Not for quote without permission

2
Motivation

• Many interesting games have more than one Nash
  equilibrium. Predicting which of these
  equilibria will be selected is perhaps the most
  important problem in behavioral game theory.
  (Camerer, 2003)

3
Games with Multiple Equilibria
The Matching Game The Matching Game The Matching Game The Matching Game The Stag Hunt Game The Stag Hunt Game The Stag Hunt Game The Stag Hunt Game
    Player 2 Player 2     Player 2 Player 2
    L R     L R
Player 1 U 80, 80 10, 10 Player 1 U 80, 80 10, 50
Player 1 D 10, 10 50,50 Player 1 D 50, 10 50,50

• Two pure strategy equilibria
• One pareto superior, one pareto inferior
• Incentives are compatible problem is
  coordination
• Stag-hunt is harder achieving pareto outcome is
  riskier

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Equilibrium Selection Criteria

• Payoff Dominance
• choose the equilibrium
• offering all players their
• highest payoff predicts UL
• Security-mindedness
• choose the strategy that
• minimizes the worst possible payoff predicts
  DR
• Risk Dominance
• choose the strategy that minimizes loses incurred
  by players as a consequence of unilaterally
  deviating from their eq. strategy predicts DR

The Stag Hunt Game The Stag Hunt Game The Stag Hunt Game The Stag Hunt Game
    Player 2 Player 2
    L R
Player 1 U 8, 8 1, 5
Player 1 D 5, 1 5,5
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An Alternative Approach Similarity Judgments in
Choice

• RX , p Y , 1-p
• SM , q N , 1-q
• Choose R (S) if it is favored in some comparisons
  and not disfavored in any, otherwise choose at
  random.

X similar / dissimilar M
p similar / dissimilar q
Y similar / dissimilar N
1-p similar / dissimilar 1-q
Favor R, Favor S, Inconclusive, Inconsequential
Favor R, Favor S, Inconclusive, Inconsequential
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Similarity Judgments The Nothing to
Gain/Nothing to Lose Effect

• R10 , .90 0 , .10
• S9 , .90 9 , .10
• R10 , .10 0 , .90
• S1 , .10 1 , .90

antg
10 x 9
.90 p .90
9 gtx 0
.10 p .10
Inconsequential
Favors S
antl
.90 p .90
10 gtx 1
.10 p .10
1 x 0
Inconsequential
Favors R
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Similarity Judgments and the Allais Paradox
a

• S3000 , .90 0 , .10
• R6000, .45 0 , .55
• S3000, .02 0 , .98
• R6000, .01 0 , .99

6000 gtx 3000 .9 gtp .45
0 x 0 .55 gtp .10
Inconclusive
Favors S
antl
6000 gtx 3000 .02 p .01
0 x 0 .99p .98
Favors R
Inconsequential
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Similarity Judgments and Intertemporal Choice

• T120, 1 month
• T225 , 2 months
• T1120, 11 month
• T1225 , 12 months

S or D? 25 gtx 20
S or D? 2 gtt 1
Inconclusive Choose either
S or D? 25 gtx 20
S or D? 12 t 11
Nothing to lose Choose T12
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Sources of prediction in Similarity based models

• Intransitivity of the similarity relation
• E.g., 20 x 17, 17 x 15 but 20 gtx 15
• Theoretically inconsequential manipulation of
  prizes, probabilities, dates of receipt may have
  consequences if they influence perceived
  similarity or dissimilarity.
• Framing of the choice
• Framing determines what is compared with what
  theoretically inconsequential changes in the
  description of the choice may influence what is
  compared with what.

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Application to Games - Preliminaries

• Assume
• Players all have the same Bernoullian utility
  function
• Let
• gtx mean is dissimilar and greater than
• A strict partial order (asymmetric and
  transitive)
• x mean is similar to
• Symmetric but not necessarily transitive (e.g.,
  20 x 15, 15 x 10 but 20 gtx10.

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Similarity Judgments in Games

• Payoffs to Player 1 h(igh) gt m(edium) gt l(ow)
• Payoffs to Player 2 t(op) gt c(enter) gt b(ottom)
• Decision process
• Do I have a dominating strategy, if so, choose
  it.
• Do I have dominating strategy in similarity if
  so, choose it.
• Does Other have dominating strategy, if so, best
  respond
• Does Other have dominating strategy in
  similarity, if so, best respond
• ?

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Similarity Judgments in the Stag-Hunt

• Player 1 (2)
• Checks for dominance
• Checks for dominance in
• similarity
• Checks for dominance for
• P2 (1), and best responds
• Checks for dominance in
• similarity for P2(1), best responds
• Chooses at random

    L R
Player 1 U 8, 8 2, 5
  D 5, 2 5, 5
   
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An Example

• If Other L and You U Y8, O8
• D 5
  2
• If Other R and You U Y2 O2
• D 5
  5
• If U(D) favored in some and not disfavored in
  any, Choose U(D), otherwise
• If L(R ) favored in some and not disfavored in
  any, best response to L(R ), otherwise random.

Favors U,D,I
Favors L,R,I
Favors U,D,I
Favors L,R,I
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The Nothing to Lose Effect and the Payoff
Dominant Eq.

• Decrease m and c.
• For Player 1
• h gtx m x l,
• Choose U ntl
• For Player 2
• t gtx c x b,
• Choose L ntl
• Outcome is payoff
• dominant UL

    L R
Player 1 U 8, 8 2, 2.1
  D 2.1, 2 2.1, 2.1
   
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The Nothing to Gain Effect and the
Security-minded Eq.

• Increase m and c
• For Player 1
• h x m gtx l,
• Choose D ntg
• For Player 2
• t x c gtx b,
• Choose R ntg
• Outcome is security-
• minded DR

    L R
Player 1 U 8, 8 2, 7.9
  D 7.9, 2 7.9, 7.9
   
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The Nothing to Gain/Nothing to Lose Effect and
Non-eq. Outcomes

• Increase m, decrease c.
• For Player 1
• h x m gtx l,
• Choose D ntg
• For Player 2
• t gtx c x b,
• Choose L ntl
• Outcome is non-equilibrium
• DL

    L R
Player 1 U 8, 8 2, 2.1
  D 7.9, 2 7.9, 2.1
   
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Predictions in the Stag Hunt
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Testing the Ntg/Ntl Effect Experiment Details

• 76 students at the University of Trento
• Experiment consisted of 3 parts, 1st of which
  involved games.
• 9 games 5 stag hunts, 3 matching pennies games,
  1 additional stag hunt (always last)
• Order otherwise randomized
• Subjects played 1 of games at end of session
  payouts between 1.20 and 8.00 euro.

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Games and Individual Results
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Results Regarding Game Outcomes
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Performance Relative to Proposed Selection
Criteria
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Games of Pure Conflict

• Players interests
• are diametrically opposed
• No equilibrium in pure
• strategies, only a mixed
• strategy

    L R
Player 1 U h, b m, t
  D l, c h, b
   
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Games of Pure Conflict and Ntg/Ntl Effects

• Player 1 compares
• high and low and
• high and middle
• Increasing m produces
• Nothing to Gain effect
• - choose U
• Player 2 compares top and bottom and bottom and
  center. Decreasing c produces Nothing to Lose
  effect choose R

    L R
Player 1 U h, b m, t
  D l, c h, b
   
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Predictions in Conflict Games
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Results
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Additional Results in Conflict Games
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Across Game Results
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Across Game Results cont.
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Other Implications - The Relativity of Similarity
Judgments
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Similarity Judgments and Framing Effects In
Choice Under Uncertainty
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Framing Effects in Games - Own First vs Other
First and non-Equilibrium Outcomes
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Framing and Question Format in Games
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Results and Implications for Quantal Response
Models
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What Would You Choose?
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Level-1 Bounded Rationality vs. Similarity
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A Speculation - the social benefit of individual
irrationality?
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A Speculation - the social benefit of individual
irrationality? (cont.)
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A Speculation - the social benefit of
non-strategic thinking and limits to learning
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Some Other Speculations and Conjectures

• Things will matter that shouldnt
• Time, recalibration and regret
• Differences in similarity perceptions and
  acrimony in negotiations

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Conclusions

• Many choice anomalies can be explained if people
  employ nothing to gain/nothing to lose
  reasoning
• The same reasoning process applied to games
  predicts
• play in coordination and conflict games and
• the successes and failures of equilibrium
  selection criteria and mixed strategy choice
• systematic differences in play as a consequence
  of theoretically inconsequential changes in the
  way strategy choices are elicited.

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References

• Camerer, C. Behavioral Game Theory Experiments
  on Strategic Interaction,
• Princeton, 2003.
• Camerer, C., Teck-Hua Ho and Juin Kuan Chong.
  Behavioral Game Theory Thinking, Learning and
  Teaching," with Teck-Hua Ho and Juin Kuan Chong.
  Forthcoming in a book edited by Steffen Huck,
  Essays in Honor of Werner Guth."
• Goerree, J. and C. Holt. Ten Little Treasures of
  Game Theory and Ten Intuitive
• Contradictions. American Economic Review. 2001.
  Vl. 91(5), pp 1402-1422.
• Haruvy, E. and D. Stahl. Deductive versus
  Inductive equilibrium selection
• experimental results. Journal of Economic
  Behavior and Organization. 2004, 53, 319-331.
• Keser, C. and B. Vogt. Why do experimental
  subjects choose an equilibrium which
• is neither risk nor payoff dominant? Cirano
  Working Paper. 2000. http//www.cirano.qc.ca/pdf/p
  ublication/2000s-34.pdf
• Leland, J. "Generalized Similarity Judgments An
  Alternative Explanation for Choice Anomalies."
  Journal of Risk and Uncertainty, 9, 1994,
  151-172.
• Leland, J. Similarity Judgments in Choice Under
  Uncertainty A Reinterpretation of Regret
  Theory. Management Science, 44(5), 1998, 1-14.
• Leland, J. Similarity Judgments and Anomalies in
  Intertemporal Choice. Economic Inquiry Vol. 40,
  No. 4, October 2002, 574-581.
• Lowenstein, G. and D. Prelec. "Anomalies in
  Intertemporal Choice Evidence and
  Interpretation." The Quarterly Journal of
  Economics, May 1992, 573-597.
• Rubinstein, A. "Similarity and Decision-making
  Under Risk (Is There a Utility Theory Resolution
  to the Allais Paradox?)." Journal of Economic
  Theory, 46, 1988, 145-153.
• Rubinstein, A. Economics and Psychology"? The
  Case of Hyperbolic Discounting,  International
  Economic Review 44, 2003, 1207-1216. 
• Standord Encycolpedia of Philosophy.
  http//plato.stanford.edu/entries/game-theory/

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Testing the Ntg/Ntl Effect Question Format
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The Problem

• existing deductive selection rules have been
  shown to do poorly in experiments (Haruvy
  Stahl, 2004)
• Should we be surprised?
• Game theory is the study strategic interactions
  among rational players..
• We know people behave irrationally in risky and
  intertemporal choice situations why would we
  expect them to do better in complex strategic
  settings?

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A Speculation - the problem with being strategic
in a non-strategic world