Nobel Laureate Thomas Schelling once proposed a solution to the problem of how a

Question

Nobel Laureate Thomas Schelling once proposed a solution to the problem of how a

kidnappee can induce his kidnapper to release him after the kidnappee has learned

the identity of the kidnapper. Let’s return to the kidnapping scenario, but instead

have the players be Guy (kidnapper) and Orlando (kidnappee). The problem is that

one would expect Guy to be inclined to kill Orlando once Orlando sees Guy’s face,

since then Orlando, if released, would be able to help the police capture Guy. The

situation is as depicted in the figure below. Guy starts off by deciding whether to

kidnap Orlando. Orlando then decides whether to reveal some incriminating details

about himself that are unknown to the rest of the world. (Perhaps Orlando stole

funds from his church or had an affair unbeknownst to his wife.) Then Guy decides

whether to kill or release Orlando. If he releases Orlando, then Orlando has to decide

whether to inform the police of his kidnapper’s identity. If he does, and if Orlando

revealed his dirty secret to Guy, Guy must then decide whether to share that secret

with the world.

Find the unique subgame perfect Nash equilibrium, and you’ll

find Schelling’s pro-posed solution.

Explanation / Answer

The game can be described as follows: To start off with, we need to know whether Guy should kidnap Orlando or not. There are two outcomes: Yes and No. If Orlando is actually kidnapped then we need to figure whether he is willing to reveal his secret or not. Again here there are two outcomes if he reveals his secret, that is, whether he should kill Orlando or not. Following from this, if Guy doesn't kill Orlando then Orlando will either inform the police of Guy or not. If he informs the police then, Guy will share Orlando's secret with the world. The nash equilibrium is when Guy doesn't kill Orlando and instead Guy reveals his secret to the world.

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