Disarmament Games
Yuan Deng
Vincent Conitzer
Duke University
Disarmament Game
• The disarmament game is
• not equivalent to the security game
• not an extension of the security game
• The disarmament game
• captures a different domain of security issues
• is highly related to commitments
Security Game
Leader-follower game
• The protector commits to a strategy first,
• then the attacker attacks
Security Game
Leader-follower game?
• One way communication
Security Game
• Two way communication?
Hostage Game
• Action of Police:
•R
• PF
• PD
:
:
:
Call the special team and Rescue directly
Pay the ransom and Follow
Pay the ransom and Don’t Follow
• Action of Terrorist:
• GR
• KR
• ST
:
:
:
Get the ransom and Run
Kill the hostage and Run
Stay
Hostage Game
R
PF
PD
GR
KR
ST
Hostage rescued
Terrorist caught
Hostage killed
Terrorist caught
Hostage killed
Terrorist caught
Hostage rescued
Hostage killed
Terrorist caught
Terrorist got away
Ransom payed
Hostage rescued
Hostage killed
Terrorist got away
Terrorist got away
Ransom payed
Nothing
happened
Nothing
happened
Hostage Game
• Police:
•
•
•
•
Hostage killed
Terrorist caught
Ransom payed
Nothing happened
:
:
:
:
-10
10
-6
-0.5
:
:
:
:
:
1
-5
2
6
-0.5
• Terrorist:
•
•
•
•
•
Hostage killed
Terrorist caught
Terrorist ran away
Ransom payed
Nothing happened
(additive valuation; unspecified valuations are simply 0)
Hostage Game
GR
KR
ST
R
10, -5
0, -4
0, -4
PF
4, 1
-10, 3
-0.5, -0.5
PD
-6, 8
-10, 3
-0.5, -0.5
Hostage Game
GR
KR
ST
R
10, -5
0, -4
0, -4
PF
4, 1
-10, 3
-0.5, -0.5
PD
-6, 8
-10, 3
-0.5, -0.5
Nash equilibria
Stackelberg equilibria (no matter who takes the lead)
Hostage Game
GR
KR
ST
R
10, -5
0, -4
0, -4
PF
4, 1
-10, 3
-0.5, -0.5
PD
-6, 8
-10, 3
-0.5, -0.5
Desired Outcome
Pareto better than the equilibrium outcome
Multiple-round (pure) commitments
Requirement:
• Once committing not to play certain strategies, the player cannot play
those strategies anymore.
• In each round
• commit to a subset of available strategies to play

• commit not to play certain strategies
Multiple-round (pure) commitments
GR
KR
ST
R
10, -5
0, -4
0, -4
PF
4, 1
-10, 3
-0.5, -0.5
PD
-6, 8
-10, 3
-0.5, -0.5
Multiple-round (pure) commitments
GR
ST
R
10, -5
0, -4
PF
4, 1
-0.5, -0.5
PD
-6, 8
-0.5, -0.5
Multiple-round (pure) commitments
GR
ST
R
10, -5
0, -4
PF
4, 1
-0.5, -0.5
PD
-6, 8
-0.5, -0.5
Incentivize Police to commit in the next round
Multiple-round (pure) commitments
GR
ST
R
10, -5
0, -4
PF
4, 1
-0.5, -0.5
PD
-6, 8
-0.5, -0.5
Multiple-round (pure) commitments
GR
ST
PF
4, 1
-0.5, -0.5
PD
-6, 8
-0.5, -0.5
Multiple-round (pure) commitments
GR
ST
PF
4, 1
-0.5, -0.5
PD
-6, 8
-0.5, -0.5
Multiple-round (pure) commitments
GR
KR
ST
R
10, -5
0, -4
0, -4
PF
4, 1
-10, 3
-0.5, -0.5
PD
-6, 8
-10, 3
-0.5, -0.5
Multiple-round (pure) commitments
GR
KR
ST
R
10, -5
0, -4
0, -4
PF
4, 1
-10, 3
-0.5, -0.5
PD
-6, 8
-10, 3
-0.5, -0.5
Fact: The desired outcome can also be achieved
if Police commits not to play PD in the first round…
Zooming out
• Consider two players who can
• communicate to each other
• make credible commitments
• Given a target outcome, can this outcome be obtained by a multipleround commitments?
Why Disarmament?
Commit not to play

Remove (Disarm) some strategies
• Consider two countries
• communicate to each other
• make credible commitments
Relation to Commitment
• One-round commitment to play (Stackelberg equilibrium)
• One-round commitment not to play
• less powerful than one round commitment to play
• (if assuming players can commit to mixed strategies)
Our model:
• Multiple-round commitments not to play
• in some cases, more powerful than one round commitment to play
• e.g. Hostage Game
Perspective from extensive-form game
The disarmament games can be formulated as an extensive-form game
• [Disarmament Stage]: sequential & deterministic
• At each decision node, one player can choose some strategies to remove
If one player is not willing to remove any strategy
• [Game Stage]: simultaneous normal-form game
• play the remaining game to get their utility
 players can play mixed strategies
Computational Problem
• Disarmament game is an extensive-form game
• Compute SPNE? By backward induction?
• In Game Stage:
• Complexity to compute NE (PPAD-complete)
• Equilibrium Selection
• Change of the structure of NE by slightly modifications
Computational Problem
• Decision problem:
• Can a specific outcome be obtained as an NE of the disarmament game?
Computational Problem
• Decision problem:
• Can a specific outcome be obtained as an NE of the disarmament game?
• If we can construct such an equilibrium, we obtain an automated way to
suggest a disarmament scheme:
• Given a disarmament objective
• Compute the Nash equilibrium  fully characterized by disarmament steps
R
PF
PD
R
PF
PD
PF
PD
GR
4, 1
-6, 8
GR
10, -5
ST
0, -4
4, 1
-6, 8
-0.5, -0.5
-0.5, -0.5
ST
-0.5, -0.5
-0.5, -0.5
GR
10, -5
4, 1
-6, 8
KR
0, -4
-10, 3
-10, 3
ST
0, -4
-0.5, -0.5
-0.5, -0.5
…
…
Computational Problem
• Decision problem:
• Can a specific outcome be obtained as an NE of the disarmament game?
• If we can construct such an equilibrium, we obtain an automated way to
suggest a disarmament scheme:
•
•
•
•
Given a disarmament objective
Compute the Nash equilibrium  fully characterized by disarmament steps
Inform both players all disarmament steps
Follow the steps without deviation (No coordination is needed!)
Computational Problem
• Decision problem:
• Can a specific outcome be obtained as an NE of the disarmament game?
# Rounds
𝑵 × 𝑲 (constant)
𝑵 × 𝑵
Constant >= 3
P
NP-complete
Unlimited
P
NP-complete
Hostage Game Review
GR
KR
ST
R
10, -5
0, -4
0, -4
PF
4, 1
-10, 3
-0.5, -0.5
PD
-6, 8
-10, 3
-0.5, -0.5
Hostage Game Review
GR
KR
R
10, -5
0, -4
PF
4, 1
-10, 3
Prisoner’s dilemma
Cooperate
Defect
Cooperate
(3,3)
(4,0)
Defect
(0,4)
(1,1)
Defect Dominates Cooperate
Prisoner’s dilemma
Cooperate
Defect
Cooperate
(3,3)
(4,0)
Defect
(0,4)
(1,1)
Defect Dominates Cooperate
Mixed Disarmament
Example : Prisoner’s Dilemma
1−𝑥
𝑥
Cooperate
Defect
1−𝑦
𝑦
Cooperate
(3,3)
(4,0)
Defect
(0,4)
(1,1)
• 𝑥, 𝑦 : maximum probability that can be placed on strategy Defect
• Defect is still a dominant strategy => place 𝑥 𝑜𝑟 𝑦 probability on Defect
• Utility :
row player
u𝑟 𝑥, 𝑦 = 3 1 − 𝑥 1 − 𝑦 + 4𝑥 1 − 𝑦 + 𝑥𝑦
column player
u𝑐 𝑥, 𝑦 = 3 1 − 𝑥 1 − 𝑦 + 4 1 − 𝑥 𝑦 + 𝑥𝑦
Example : Prisoner’s Dilemma
1−𝑥
𝑥
Cooperate
Defect
1−𝑦
𝑦
Cooperate
Defect
(3,3)
(4,0)
(0,4)
(1,1)
•
u𝑟 𝑥, 𝑦 = 3 1 − 𝑥 1 − 𝑦 + 4𝑥 1 − 𝑦 + 𝑥𝑦 ≤ 3
•
u𝑐 𝑥, 𝑦 = 3 1 − 𝑥 1 − 𝑦 + 4 1 − 𝑥 𝑦 + 𝑥𝑦 ≤ 3
u𝑟 𝑥, 𝑦 ≤ 3
u𝑐 𝑥, 𝑦 ≤ 3
𝑦
𝑥
Cooperate
Defect (x)
Cooperate
(3,3)
(12,0)
Defect (y)
(0,12)
(1,1)
u𝑟 𝑥, 𝑦 ≤ 4
u𝑐 𝑥, 𝑦 ≤ 4
𝑦
𝑥
u𝑟 = 𝑢𝑐 = 4
Folk Theorem
• In repeated games with infinite rounds, any desired utilities
can be obtained as an equilibrium.
• Desired: (1) feasible: can be obtained by a mixed strategy profile
(2) enforceable: larger than the utilities they can guarantee themselves
no matter how the other player plays
Cooperate
Defect
Cooperate
Defect
(3,3)
(4,0)
(0,4)
(1,1)
Folk Theorem
• In repeated games with infinite rounds, any desired utilities
can be obtained as an equilibrium.
• Desired: (1) feasible: can be obtained by a mixed strategy profile
(2) enforceable: larger than the utilities they can guarantee themselves
no matter how the other player plays
• In mixed disarmament games, for any desired utilities, there
exists an equilibrium achieving at least these utilities.
Folk Theorem without repetition! 
Folk Theorem (# of steps)
• In mixed disarmament games, for any desired utilities, there
exists an equilibrium achieving at least these utilities
• For any desired utilities, we can construct an 𝑛𝜖-(additive)
approximate Nash equilibrium with almost the same outcome
(differed by at most 𝑛𝜖) and 𝑂(1/𝜖) 𝑦disarmament steps.
 𝑛 is the number of strategies
𝑥
Back to Disarmament
• Based on Folk Theorem
• Two countries only need to negotiate about outcome utilities, that are,
• feasible
• enforceable
• Fair? Social optimal?...
• No worry about how to achieve it
Conclusion & Future works
• Summary
• Formulate disarmament games
• Investigate computational complexity for the pure case
• Show a folk theorem for the mixed case
• Future work
• Game representation other than normal-form games (e.g. Bayesian game)
• Restricted class of games
• Applications?