1. Yuan Deng Vincent Conitzer Duke University
Disarmament Games
Yuan Deng Vincent Conitzer
Duke University

2. Disarmament Game The disarmament game is The disarmament game
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

3. Security Game Leader-follower game
The protector commits to a strategy first,
then the attacker attacks

4. Security Game
Leader-follower game?
One way communication

5. Security Game
Two way communication?

6. Hostage Game Action of Police: Action of Terrorist:
R : Call the special team and Rescue directly
PF : Pay the ransom and Follow
PD : Pay the ransom and Don’t Follow
Action of Terrorist:
GR : Get the ransom and Run
KR : Kill the hostage and Run
ST : Stay

7. Hostage Game GR KR ST R Hostage rescued Terrorist caught
Hostage killed PF
Ransom payed
Terrorist got away
Nothing happened PD

8. Hostage Game Police: Terrorist: Hostage killed : -10
Terrorist caught : 10
Ransom payed : -6
Nothing happened : -0.5
Terrorist:
Hostage killed : 1
Terrorist caught : -5
Terrorist ran away : 2
Ransom payed : 6
(additive valuation; unspecified valuations are simply 0)

9. Hostage Game GR KR ST R 10, -5 0, -4 PF 4, 1 -10, 3 -0.5, -0.5 PD
-6, 8

10. Hostage Game GR KR ST R 10, -5 0, -4 PF 4, 1 -10, 3 -0.5, -0.5 PD
-6, 8
Nash equilibria
Stackelberg equilibria (no matter who takes the lead)

11. Hostage Game GR KR ST R 10, -5 0, -4 PF 4, 1 -10, 3 -0.5, -0.5 PD
-6, 8
Desired Outcome
Pareto better than the equilibrium outcome

12. 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

13. Multiple-round (pure) commitmentsGR KR ST R
10, -5
0, -4 PF
4, 1
-10, 3
-0.5, -0.5 PD
-6, 8

14. Multiple-round (pure) commitmentsGR ST R
10, -5
0, -4 PF
4, 1
-0.5, -0.5 PD
-6, 8

15. Multiple-round (pure) commitmentsGR ST R
10, -5
0, -4 PF
4, 1
-0.5, -0.5 PD
-6, 8
Incentivize Police to commit in the next round

16. Multiple-round (pure) commitmentsGR ST R
10, -5
0, -4 PF
4, 1
-0.5, -0.5 PD
-6, 8

17. Multiple-round (pure) commitmentsGR ST PF
4, 1
-0.5, -0.5 PD
-6, 8

18. Multiple-round (pure) commitmentsGR ST PF
4, 1
-0.5, -0.5 PD
-6, 8

19. Multiple-round (pure) commitmentsGR KR ST R
10, -5
0, -4 PF
4, 1
-10, 3
-0.5, -0.5 PD
-6, 8

20. Multiple-round (pure) commitmentsGR KR ST R
10, -5
0, -4 PF
4, 1
-10, 3
-0.5, -0.5 PD
-6, 8
Fact: The desired outcome can also be achieved
if Police commits not to play PD in the first round…

21. 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 multiple- round commitments?

22. Why Disarmament? Commit not to play  Remove (Disarm) some strategies
Consider two countries
communicate to each other
make credible commitments

23. 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

24. 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

25. 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

26. Computational Problem
Decision problem:
Can a specific outcome be obtained as an NE of the disarmament game?

27. 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

28. … … GR KR ST R 10, -5 0, -4 PF 4, 1 -10, 3 -0.5, -0.5 PD -6, 8 GR ST R

29. 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!)

30. 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

31. Hostage Game Review GR KR ST R 10, -5 0, -4 PF 4, 1 -10, 3 -0.5, -0.5PD
-6, 8

32. Hostage Game Review GR KR R
10, -5
0, -4 PF
4, 1
-10, 3

33. Prisoner’s dilemma Cooperate Defect (3,3) (0,4) (4,0) (1,1)
Defect Dominates Cooperate

34. Prisoner’s dilemma Cooperate Defect (3,3) (0,4) (4,0) (1,1)
Defect Dominates Cooperate

35. Mixed Disarmament

36. Example : Prisoner’s Dilemma
1−𝑦 𝑦
Cooperate
Defect
(3,3)
(0,4)
(4,0)
(1,1)
1−𝑥 𝑥
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 u 𝑟 𝑥,𝑦 = 3 1−𝑥 1−𝑦 +4𝑥 1−𝑦 +𝑥𝑦
column player u 𝑐 𝑥,𝑦 = 3 1−𝑥 1−𝑦 +4 1−𝑥 𝑦 +𝑥𝑦

37. Example : Prisoner’s Dilemma
1−𝑦 𝑦
Cooperate
Defect
(3,3)
(0,4)
(4,0)
(1,1)
1−𝑥
u 𝑟 𝑥,𝑦 ≤3
u 𝑐 𝑥,𝑦 ≤3 𝑥 𝑦
u 𝑟 𝑥,𝑦 = 3 1−𝑥 1−𝑦 +4𝑥 1−𝑦 +𝑥𝑦≤3
u 𝑐 𝑥,𝑦 = 3 1−𝑥 1−𝑦 +4 1−𝑥 𝑦 +𝑥𝑦≤3 𝑥

38. Cooperate Defect (y) Defect (x)
(3,3)
(0,12)
Defect (x)
(12,0)
(1,1)
u 𝑟 𝑥,𝑦 ≤4
u 𝑐 𝑥,𝑦 ≤4 𝑦 𝑥
u 𝑟 = 𝑢 𝑐 =4

39. 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
(3,3)
(0,4)
(4,0)
(1,1)

40. Folk Theorem without repetition! 
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! 

41. 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) disarmament steps.
𝑛 is the number of strategies 𝑦 𝑥

42. 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

43. 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?