metagame strategies of nation-states, with application to cross-strait relations alex chavez and jun...
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Metagame Strategies of Nation-States, with Application
to Cross-Strait Relations
Alex Chavez and Jun Zhang*
Dept. of Psychology, University of Michigan
*AFOSR
Standard solution concept• Nash equilibrium (NE):
• NE often fails descriptively.– Common knowledge of
rationality.• Limited # of steps of iterated
thinking (Camerer, 2003).
– Utility misspecifications.
• Altruism, inequality aversion (Fehr & Schmidt, 1998), social norms (Bicchieri, 2006).
– Strategy space?
Cooperate Defect
Cooperate 3, 3 0, 5
Defect 5, 0 1, 1
Why?
MetagamesMetagames describe situations where players
recursively predict each other’s conditional strategies.
• Base game: where P = set of players, S = strategy
space, π = payoff functions.• Metagame: Iteratively replace Si with
• Each metagame is identified by its title, the order in which the Si* are constructed.
• E.g., some metagames for P = {1, 2} are:
Metagames• Example: 21Γ for Γ = Prisoner’s Dilemma• Player 1:
Level-1
Level-2
• Player 2:
f1
If Player 2 Then
Cooperate Cooperate
Defect Cooperate
f3
If Player 2 Then
Cooperate Defect
Defect Cooperate
f2
If Player 2 Then
Cooperate Cooperate
Defect Defect
f4
If Player 2 Then
Cooperate Defect
Defect Defect
g1
If Player 1 Then
f1 Cooperate
f2 Cooperate
f3 Cooperate
f4 Cooperate
g2
If Player 1 Then
f1 Cooperate
f2 Defect
f3 Cooperate
f4 Cooperate
g16
If Player 1 Then
f1 Defect
f2 Defect
f3 Defect
f4 Defect
. . .
Metagames• Metagames strategies resolve as base game strategies.
f2
If Player 2 Then
Cooperate Cooperate
Defect Defect
g2
If Player 1 Then
f1 Cooperate
f2 Defect
f3 Cooperate
f4 Cooperate
Γ
(c,c) (c,d)
(d,c) (d,d)
21Γ
Metagames• Metagames strategies resolve as base game strategies.
f2
If Player 2 Then
Cooperate Cooperate
Defect Defect
g2
If Player 1 Then
f1 Cooperate
f2 Defect
f3 Cooperate
f4 Cooperate
Γ
(c,c) (c,d)
(d,c) (d,d)
21Γ
Metagames• Metagames strategies resolve as base game strategies.
f2
If Player 2 Then
Cooperate Cooperate
Defect Defect
g2
If Player 1 Then
f1 Cooperate
f2 Defect
f3 Cooperate
f4 Cooperate
Γ
(c,c) (c,d)
(d,c) (d,d)
21Γ
Metagames• Metagames strategies resolve as base game strategies.
f2
If Player 2 Then
Cooperate Cooperate
Defect Defect
g2
If Player 1 Then
f1 Cooperate
f2 Defect
f3 Cooperate
f4 Cooperate
Γ
(c,c) (c,d)
(d,c) (d,d)
21Γ
Metagames• Metagames strategies resolve as base game strategies.
Γ
(c,c) (c,d)
(d,c) (d,d)
21Γ
Metagames• Metagames strategies resolve as base game strategies.
Γ
(c,c) (c,d)
(d,c) (d,d)
21Γ
Metagames• Metagames strategies resolve as base game strategies.
Γ
(c,c) (c,d)
(d,c) (d,d)
21Γ
Metagames• Metagames strategies resolve as base game strategies.
• Resolution is easy. Finding Nash equilibria is not.– E.g., 24 x 232 x 2512 outcomes in the 3-player game we study.
• Luckily, it is easy to find metaequilibria, outcomes in the base game which the Nash equilibria in the metagame game project to.
Γ
(c,c) (c,d)
(d,c) (d,d)
21Γ
Three useful theorems• (Identification). Howard (1971) provides a theorem for
identifying the set of all metaequilibria.– Requires optimization over certain strategy subspaces of the
base game.
• (Reducilibility). Repetitions in the title may be deleted.
• (Nestedness). Metaequilibria are nested in larger titles.
Application: cross-Strait relations
• 1949 Communist party take power of mainland China after civil war with nationalists, who setup a government in Taiwan.
• 1979 U.S. recognition of communist China and passage of Taiwan Relations Act, which protects Taiwan against Chinese attack
• Recent years:• Taiwan indicates desire of official independence
from mainland China.• China threatens to use force to prevent this.• The U.S. may have a pro-Taiwan or pro-China
stance.
Taiwan Taiwan
No Ind. Ind. No Ind. Ind.
ChinaNo War A B E F
War C D G H
U.S.: Support U.S.: No support
(A) Status quo (E) Isolated Taiwan without independence
(B) U.S.-recognized independent Taiwan
(F) U.S.-unrecognized independent Taiwan
(C) Taiwan gives in despite U.S. support
(G) Unification without resistance
(D) All-out war (H) Unification w/resistance
Results• G, forceful unification without
resistance, is a metaequilibrium in every metagame by the nesting property.
• The status quo, A, is a metaequilibrium in certain level-2 metagames and in all level-3 metagames.
Results• Brute force -> all Nash equilibria of cΓ.• E.g., for G:
– Taiwan does not declare independence,– The U.S. does not support Taiwan, and – China threatens to go to war if either Taiwan or the
U.S. unilaterally changes strategies.
Summary and Future Directions• Metagames
– Applied to multinational conflict.– Useful for highly sophisticated players.
• Open questions– Robustness to payoff assumptions– Computation of Nash equilibria– Real challenge: qualitatively describing the
many Nash equilibria associated with one metaequilibrium
• Thanks. questions?
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