In this clip from the CBS Show Numb3rs, in which a mathematician (Charlie) helps his FBI agent brother (Don) fight crime using his number theory-driven insights, we see Charlie create a prisoners’ dilemma situation to get one of the suspects to talk. Although the show is entertaining and they often reference real mathematical theorems and properties, some mathematicians who have tried many of Charlie’s approaches have found them to not be applicable after all. In this spirit of checking Charlie’s work, I decided to take a clip from the show in which Charlie references game theory (of which there are several) and see if what he says holds water.
First, in his explanation to his brother about what game theory is, he says game theory is the process by which you try to achieve “optimal outcome” from a complex situation – that part sounds all right. Then Charlie provides him with an example: 2 prisoners and how much time they would each serve if they talk or don’t talk and if the other prisoner talks or doesn’t talk. I’ve written up the game set-up below and included an identification of dominant strategies, etc. according to what we’ve learned in class. Looks like Charlie’s example is a good one. Moving forward…
Don presents the case: three men, F, G, and W, have stolen a vehicle filled with civilians that must be found as soon as possible. As they discuss strategy, Don shares the go-to move of keeping the suspects isolated from one another while simultaneously trying to pit them against one another. As we learned, the assumptions behind a game are as follows: (1) payoff summarizes everything a player cares about (2) each player knows everything about the structure of the game: who players are, strategies available to all, payoffs for each player/strategy (3) every player is rational: wants to maximize payoff and succeeds in doing so. According to these assumptions, Don’s own prisoner’s dilemma met all the assumptions of a coordination game – except for the last part of the second assumption: because they were being kept separate, the men did not know about payoffs for each player, they only knew their own. Charlie suggested bringing them together, which rectified this situation. He also looked at a collection of factors such as existing criminal record and connections to family “on the outside” to determine what each man was risking by refusing to act on his dominant strategy (meaning by still choosing to not talk). This way, everyone knew how much was at stake for their accomplices and could consider how that would affect their strategy.
By taking Charlie’s advice, Don changed the dynamic in the room, among the men, and between himself and the men. The following illustration shows the impact of Don asserting himself into the men’s network as the only node connected to all 3 nodes. G had been the center of the chain since he was the ringleader, but now Don boasts the most powerful position because he is connected to all 3 other nodes and could get what he wants from any of them.
In this particular clip, it looks like Charlie’s theory actual worked out flawlessly. Although the game that Charlie proposed for these suspects was viable, the game assumes that every player was rational, which is not always the case with real-live humans. The assumptions behind a game also include that the payoff (or in this case, risk assessment) summarizes everything a player cares about. Since the men’s risk assessments were created by Charlie through the use of factors he anticipated held value for the men, they were automatically a “best guess” because they were not made with complete certainty that these processes summarized everything that was important to these men and impacted their going to prison. Despite these shortcomings however, the idea behind the Charlie’s game theory-inspired strategy holds true: you can influence people better if they see firsthand what they’re up against and a rational actor will adopt their optimal strategy to secure themselves a much greater likelihood of a favorable (or comparatively favorable) outcome.