Three-Player Prisoner’s Dilemma
Consider a game of Prisoner’s Dilemma. Only instead of two criminals, the FBI must interrogate three criminals. What would the payoff matrix look like for this game? What would be the rules? What would be the Dominate Strategy for each player?
Suppose the payoff matrix for a three-player game of Prisoner’s Dilemma is 3-D, then it would look like a cube, made up of 8 smaller cubes. Each cube represents a different outcome. Listed below are the outcomes, C for to confess and NC for not to confess.
(Player 1, Player 2, Player 3)
(C, C, C)
(C, NC, C)
(C, C, NC)
(C, NC, NC)
(NC, C, C)
(NC, NC, C)
(NC, C, NC)
(NC, NC, NC)
Suppose there are some rules. If all three players confess, each player gets 8 months in jail. If all three players refuse to confess, each player gets 4 months. If only one player confesses, he/she walks, and the other two gets 12 months each. Finally, if two players confess, they get 6 months each, and the third player gets 12 months. Listed below are the outcomes with appropriate jail times for each player.
(Player 1, Player 2, Player 3) = (#, #, # )
(C, C, C) = (8, 8, 8 )
(C, NC, C) = (6, 12, 6 )
(C, C, NC) = (6, 6, 12 )
(C, NC, NC) = (0, 12, 12 )
(NC, C, C) = (12, 6, 6 )
(NC, NC, C) = (12, 12, 0 )
(NC, C, NC) = (12, 0, 12 )
(NC, NC, NC) = (4, 4, 4 )
Let’s analysis the outcomes from Player 1’s perspective: if Player 2 and Player 3 confesses, Player 1 should confess, 12>8; if Player 2 confesses and Player 3 refuses to confess, Player 1 should confess, 12>6; if Player 2 refuses to confess and Player 3 confesses, Player 1 should confess, 12>6; if Player 2 and Player 3 refuse to confess, Player 1 should confess, 4>0. Thus, to confess is a Dominate Strategy for Player 1, this process works the same way for Player 2 and Player 3.
The above discussion applies only to a game in which all players perceive a specific length of jail time as equally bad. For example, in an episode of Numb3rs, “Dirty Bomb,” mathematician Charlie Epps used Game Theory and Risk Analysis to aid FBI agent Don Epps in interrogating three criminals. Charlie integrated Risk Analysis into the Prisoner’s Dilemma. He argued that one criminal may have more to lose by going to jail than anther. Thus, assigning months alone to the payoff matrix may be misleading. Charlie performed a Risk Analysis for each criminal and derived a factor for each criminal: (Player 1 = 7.9, Player 2 = 14.9, Player 3 = 26.4). If all three criminals go to jail for the same length of time, Player 3 has more to lose than the others. Listed below are the outcomes with appropriate jail times multiplied by risk factors for each player.
(Player 1, Player 2, Player 3) = (#, #, # )
(C, C, C) = (63.2, 119.2, 211.2 )
(C, NC, C) = (47.4, 178.8, 158.4 )
(C, C, NC) = (47.4, 89.4, 316.8 )
(C, NC, NC) = (0, 178.8, 316.8 )
(NC, C, C) = (94.8, 89.4, 158.4 )
(NC, NC, C) = (94.8, 178.8, 0 )
(NC, C, NC) = (94.8, 0, 316.8 )
(NC, NC, NC) = (31.6, 59.6, 105.6 )
Dominate Strategy:
Player 1: C – Player 2: C → Player 3: C
Player 1: C – Player 2: NC → Player 3: C
Player 1: NC – Player 2: C → Player 3: C
Player 1: NC – Player 2: NC → Player 3: C
Player 3’s Dominate Strategy = C
Player 1: C – Player 3: C → Player 2: C
Player 1: C – Player 3: NC → Player 2: C
Player 1: NC – Player 3: C → Player 2: C
Player 1: NC – Player 3: NC → Player 2: C
Player 2’s Dominate Strategy = C
Player 3: C – Player 2: C → Player 1: C
Player 3: C – Player 2: NC → Player 1: C
Player 3: NC – Player 2: C → Player 1: C
Player 3: NC – Player 2: NC → Player 1: C
Player 1’s Dominate Strategy = C
The Dominate Strategy for each player is still to confess! In Numb3rs, the FBI agent put all three criminals into the same room. Then, the mathematician presented each criminal’s risk factor. Player 1 remained indifferent, Player 2 hesitated, and Player 3 confessed almost immediately after. The actual outcome of the game is (NC, NC, C). Even though each player’s Dominate Strategy is to confess, it should be considered that some players may be more motivated to confess than others. And if timing matters, the criminal with the most to lose may confess before the others.
Here’s the relevant clip from the episode:
http://www.youtube.com/watch?v=I0pwjoBanLk
Cornell’s math department has summarized the applied mathematics components of each episode in this website:
http://www.math.cornell.edu/~numb3rs/
What if we make it a repeated game? Can that lead to a NE of (NC, NC, NC)?