The game theory behind the new NBC show called “Take it All”

There’s a new NBC game show called Take it All which recreates a well-studied problem in game theory called the Prisoner’s dilemma.   According to the Prisoner’s dilemma Wikipedia article, a “classic” example of this game is as follows:

“Two men are arrested, but the police do not have enough information for a conviction. The police separate the two men, and offer both the same deal: if one testifies against his partner (defects/betrays), and the other remains silent (cooperates with/assists his partner), the betrayer goes free and the one that remains silent gets a one-year sentence. If both remain silent, both are sentenced to only one month in jail on a minor charge. If each ‘rats out’ the other, each receives a three-month sentence. Each prisoner must choose either to betray or remain silent; the decision of each is kept secret from his partner until the sentence is announced. What should they do?”

This game is called the Prisoner’s dilemma because the solution to the game involves joint betrayal rather than joint cooperation, even though joint cooperation is the better outcome for both.  To see this, consider the following “payoffs” (in terms of prison time) that Prisoner 1 and Prisoner 2 associate with the strategies “Betray” and “Keep Silent”.

 

Prisoner 2

Prisoner 1

Betray

Keep Silent

Betray

A) 3 months in jail, 3 months in jail

C) Go free, 1 year in jail

Keep Silent

B) 1 year in jail, Go free

D) 1 month in jail, 1 month in jail

Suppose Prisoner 2 decides to betray Prisoner 1.  Under the “Prisoner 2 Betrays Prisoner 1” scenario, Prisoner 1 will only have to spend 3 months in jail if he betrays Prisoner 2 (corresponding to cell A above), and a full year in jail if he keeps silent (corresponding to cell B above).  Now suppose Prisoner 2 decides to keep silent.  Under the “Prisoner 2 keeps silent” scenario, Prisoner 1 goes free if he betrays Prisoner 2 (corresponding to cell C above), and spends 1 month in jail if he keeps silent (corresponding to cell D above).  So no matter what Prisoner 2 does, Prisoner 1 always spends less time in jail if he betrays Prisoner 2.  By symmetry, no matter what Prisoner 1 does, Prisoner 2 always spends less time in jail if he betrays Prisoner 1.   Thus the dilemma…

Now, let’s consider how Take it All recreates the Prisoner’s dilemma.  The show begins with five contestants playing a first round, four contestants playing a second round, and three contestants playing a third round.  The final two contestants after round three advance to the “Prize Fight” which mimics the Prisoner’s dilemma game shown above.  In the Prize Fight, the strategy pair for each contestant is “Keep Mine” (i.e., keep my winnings from Rounds 1–3) and “Take it All” (i.e., keep my winnings from Rounds 1–3 and also take my opponent’s winnings from Rounds 1–3).  If both contestants choose to “Keep Mine,” they will each keep the prizes they have won in the prior rounds (this is analogous to ending up in cell D in the Prisoner’s dilemma payoff matrix shown above). If one contestant chooses “Keep Mine” and the other chooses to “Take it All,” the contestant that chose “Take it All” will go home with all the prizes — theirs and their opponents (this is analogous to ending up in either cell B or cell C D in the Prisoner’s dilemma payoff matrix).  But if both choose “Take it All,” they both go home with nothing (this is analogous to ending up in cell A in the Prisoner’s dilemma payoff matrix).  If one were to play this game purely on the basis of self-interested action, then the obvious strategy would be to always “Take it All”, in which case NBC doesn’t have to pay out at all.  However, the interesting (and somewhat creepy and provocative twist) here is that the show’s host (Howie Mandel) gives both contestants time to try to convince each other that they won’t betray each other.  Thus, the “suspense” is whether one’s opponent follows through on his or her verbal “promise” to “Keep Mine”.

Now consider what happened earlier this week on a “Take it All” episode:

From a business perspective, I think that it is quite clever to produce a game show based upon a prisoner’s dilemma game.  The production costs for a show like this are nominal, and most of the time the production company won’t have to pay out a jackpot.  It doesn’t take a game theory expert to figure out that the incentive structure of the game is incompatible with a (Keep Mine, Keep Mine) outcome.  The  likely outcome is (Take It All, Take It All), in which case no prize is paid.  Occasionally one can expect a (Keep Mine, Take It All) outcome, as was the case earlier this week.  This also works to the NBC’s benefit; the “drama” of the game creates a buzz and possibly more viewership (thus bringing in more revenue from commercials).  In all likelihood, NBC also manages the risk of a (Keep Mine, Take It All) outcome by purchasing an insurance policy covering the cost of the jackpot from Lloyds of London…

Let me make one final point about Prisoners Dilemmas and television shows.  The next time you watch a Law and Order re-run, keep in mind that virtually every Law and Order episode is organized around some variation of a Prisoners Dilemma game.  The downside of realizing this is that this makes Law and Order much less interesting to watch, since you can often deduce what will likely happen just a few minutes into each episode!

 

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