Question

Solomon’s Mechanism: An object belongs to one of two people - 1 or 2, each of whom claims ownership. The value of the object is H to the owner and L to the 1 other individual, where H > L > 0. King Solomon orders the two people to play the following game. Randomly, by tossing up a coin, the players are assigned to be first mover (M1) and second mover (M2). M1 starts and has to declare either mine or hers. If she says hers, the game is over and M2 gets the object. If she says mine, then M2 gets to move. M2 has to say either hers, in which case the object is given to M1, or mine, in which case M2 gets the object and pays K to King Solomon, with H > K > L, and M1 pays a small amount ε > 0 to King Solomon.

2.1 [10 points] Assume that the true owner of the object is player 1. Draw the extensive form expression of the game.

2.2 [10 points] Explain, using the concept of SPNE, why the outcome of this procedure is that the owner gets the object without paying anything.

Answer 1

(a) Randomly, by tossing up a coin, the players are assigned to be first mover (M1) and second mover (M2).

(b) Assuming the owner is player 1.

Now if M1 is player 1, solving the game by backward induction. M2(player 2) will be better by saying that the obect is hers because saying hers will derive 0 payoff whereas claiming mine will provide L-K payoff(which is negative). So, in this case M1 will get the object without paying anything.

If M2 is player 1, then she will be better off by claiming mine because H-K is positive. On the contrary if she says hers, she will get 0 payoff. Now, M1(player 2) knows that M2 will claim mine. In this situation, she will be betteroff if she says hers. This is because claiming mine will yield M2 -e payoff whereas saying hers will give 0 payoff. Therefore, again the owner will get the product without anything for it.