Dear GameMaster, If it is not too much trouble, could you please roll an ordinary six sided die? If it comes up 1 or 2, then please indicate that fact privately to Mr. Loeb. If it comes up 5 or 6, then please indicate that fact privately to Mr. Shoham. If it comes up 3 or 4, then please take no special action. We are trying to pick a random strategy (we must keep those other players guessing!), and we don't have a die handy. Thanks in advance. Yours, Danny Loeb Dan ShohamWhat is going on here?
Why do these two players (actually game theorists) ask their GM to roll a die when they could very well do so themselves?
One possibility is that they are deciding if a given center will go to Loeb or to Shoham, and they do not trust each other to flip the coin fairly without the presence of the other player. (Actually there are methods of simulating a coin flip between two separated parties without the intervention of a neutral third party.) Still, if they really wanted to use a neutral third party, why not use the final digit of the Dow Jones index as a "die"?
A hint as to why they couldn't do this lies in their strange instructions to the GM as to how to report the results. They clearly want one party to be in doubt as to the result of the die roll. If they had agreed to use some publically available source of information, this would be impossible.
Well, maybe Loeb has agreed to support Shoham. Loeb has two units available for the support. Whichever unit Loeb uses will expose him to a stab by Shoham in a different way, so he doesn't want Shoham to know which unit will actually be making the support.
That is possible, but quite unlikely, since if the secret decision only involves Loeb's unit, then why doesn't he make up his mind "randomly" by himself? Furthermore, why does Shoham get information in certain cases?
What really is going on is that both players are being given access to their own "random variable." These random variables take two values:
However, we must note that these random variables are not "independent." In particular, there is no chance that both players will be contacted by the GM.
Here is what is really going on. To keep things, simple let's assume that both players have only two options: "conservative" and "ambitious." (I'll avoid actually drawing a Diplomacy gameboard, since any real position is bound to be more complicated and will divert your attention from my main point.)
If both allies chose to be "conservative" then they both get a good result. Let's say the resulting position is worth "six" in some units to both players. If both players are "ambitious," then the plan would fail, and both players get "zero." If one player is "ambitious" while the other is "conservative," then the "ambitious" player gets a great position worth "seven" and the "conservative" player gets a poor position worth only "two."
+-------+ | Loeb | +---+---+ | C | A | +--------+---+---+---+ | | |\ 6|\ 7| | | C | \ | \ | | | |6 \|2 \| | Shoham +---+---+---+ | | |\ 2|\ 0| | | A | \ | \ | | | |7 \|0 \| +--------+---+---+---+The Nobel Prize winning mathematician John Nash showed that any such game has (at least one) equilibrium position from which no player has an incentive to change his moves. In this example, the result (Loeb=ambitious, Shoham=conservative) is a Nash equilibrium, since Loeb and Shoham would lose 1 and 2 points respectively if either changed his mind. Similarly, (Loeb=conservative, Shoham=ambitious) is a Nash equilibrium.
In actual play, each player will try to impose the equilibrium that favors him by trying to argue to his ally that he is committed to being ambitious, has already sent in his moves, and will have no chance to change them. The risk, of course, if both players do that is that they will both end up with zero. In any case, Nash Equilibria play leads only to a "total gain" of "nine" for the alliance.
Now, in the situation as described, Loeb and Shoham have agreed to only be ambitious if he is contacted by the GM. This agreement will result in (conservative, conservative), (conservative, ambitious), and (ambitious, conservative) each one-third of the time. The average expected payoff for the alliance is now "ten."
However, is the agreement stable? That is, do both players have an incentive to keep their agreements in all circumstances? After all, if we were to consider all agreements regardless of their stability, then an outcome of "twelve" would come from playing (conservative, conservative).
If the GM contacts Loeb, then Loeb knows that Shoham will not be contacted and will play "conservative." Loeb can ensure that highest payoff possible for himself by sticking to the prearranged strategy and play "ambitious."
If the GM does not contact Loeb, then the GM may have contacted Shoham or he might not have. From Loeb's point of view, either may have occurred with an a postiori probability of one-half. By playing "conservative" (as per the agreement), Loeb will gain "six" half of the time, and "two" half of the time, leading to an expected gain of "four." By breaking the agreement and playing "ambitious," Loeb will gain "seven" half of the time, and "zero" half of the time, leading to an expected gain of "3.5."
Thus, in neither case should Loeb break his agreement. Shoham will prove to be trustworthy for similar reasons.
Each player has an information set that is different from but related to the information set of the other player. By not knowing everything that your ally knows, a certain stability is achieved that allows the alliance to perform better.
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