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-1Level-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?

1

2 3

2131

12

32 13

23

321231

312

132 213

123