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. Why?
– Common knowledge of
rationality.
• Limited # of steps of iterated
thinking (Camerer, 2003).
Cooperate
Defect
Cooperate
3, 3
0, 5
Defect
5, 0
1, 1
– Utility misspecifications.
• Altruism, inequality aversion
(Fehr & Schmidt, 1998),
social norms (Bicchieri,
2006).
– Strategy space?
Metagames
Metagames 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:
f1
If Player 2
Then
f2
Cooperate Cooperate
If Player 2 Then
Defect
Cooperate
Cooperate Cooperate
Level-1
Level-2
Defect
Defect
f3
If Player 2 Then
f4
Cooperate Defect
If Player 2 Then
Defect
Cooperate
Cooperate Defect
Defect
Defect
• Player 2:
g1
g2
If Player 1 Then
If Player 1 Then
f1
Cooperate
f1
Cooperate
f2
Cooperate
f2
Defect
f3
Cooperate
f3
f4
Cooperate
f4
g16
If Player 1 Then
...
f1
Defect
f2
Defect
Cooperate
f3
Defect
Cooperate
f4
Defect
Metagames
• Metagames strategies resolve as base game strategies.
Γ
21Γ
(c,c) (c,d)
(d,c) (d,d)
g2
f2
If Player 1 Then
If Player 2
Then
f1
Cooperate
Cooperate Cooperate
f2
Defect
Defect
f3
Cooperate
f4
Cooperate
Defect
Metagames
• Metagames strategies resolve as base game strategies.
Γ
21Γ
(c,c) (c,d)
(d,c) (d,d)
g2
f2
If Player 1 Then
If Player 2
Then
f1
Cooperate
Cooperate Cooperate
f2
Defect
Defect
f3
Cooperate
f4
Cooperate
Defect
Metagames
• Metagames strategies resolve as base game strategies.
Γ
21Γ
(c,c) (c,d)
(d,c) (d,d)
g2
f2
If Player 1 Then
If Player 2
Then
f1
Cooperate
Cooperate Cooperate
f2
Defect
Defect
f3
Cooperate
f4
Cooperate
Defect
Metagames
• Metagames strategies resolve as base game strategies.
Γ
21Γ
(c,c) (c,d)
(d,c) (d,d)
g2
f2
If Player 1 Then
If Player 2
Then
f1
Cooperate
Cooperate Cooperate
f2
Defect
Defect
f3
Cooperate
f4
Cooperate
Defect
Metagames
• Metagames strategies resolve as base game strategies.
Γ
21Γ
(c,c) (c,d)
(d,c) (d,d)
Metagames
• Metagames strategies resolve as base game strategies.
Γ
21Γ
(c,c) (c,d)
(d,c) (d,d)
Metagames
• Metagames strategies resolve as base game strategies.
Γ
21Γ
(c,c) (c,d)
(d,c) (d,d)
Metagames
• Metagames strategies resolve as base game strategies.
Γ
21Γ
(c,c) (c,d)
(d,c) (d,d)
• 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.
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
China
Taiwan
No Ind.
Ind.
No Ind.
Ind.
No War
A
B
E
F
War
C
D
G
H
U.S.: Support
(A)
Status quo
(B)
U.S.-recognized
independent Taiwan
Taiwan gives in
despite U.S. support
All-out war
(C)
(D)
U.S.: No support
(E) Isolated Taiwan without
independence
(F) U.S.-unrecognized
independent Taiwan
(G) Unification without
resistance
(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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