metagame strategies of nation-states, with application to cross-strait relations alex chavez and jun...

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


Γ

(c,c) (c,d)

(d,c) (d,d)

21Γ


• 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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metagame strategies of nation-states, with application to cross-strait relations alex chavez and jun...

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