GAME THEORY: SOME EXAMPLES:Here are a few
examples of what the author Siegfried uses to illustrate Von
Neumann's game theory in his book 'A Beautiful Math'.
The first is from real life and has a trivial outcome
ie a single best strategy fort each player no matter what the other player does.
During WW2
General George Kenney was in charge of the allied air forces in the southwest Pacific. During the battle of the Bismark Sea he knew that the
Japanese would be sending a convoy of supply ships to New Guinea. The Allies wanted to bomb this convoy and could get in three days of possible bombing if the convoy too either of two possible routes, either north or south of New Britain. The
complication was that the northern route would be stormy for one of these days, leaving only two clear days for bombing.
Kenney could send his
reconnaissance planes either north or south but not both (too few planes to cover both routes I guess). If they sighted the convoy the attack planes would follow. If not the attack force would head in the opposite direction the next day by a process of elimination.
I think you can see where the complications in this scenario lead, complications not mentioned by Siegfried and apparently not by the many textbooks that use this example. This first is why not send the attack planes in the opposite direction if the convoy is not sighted in due time ? The second is that one has to assume that
reconnaissance is
100% effective. If the planes say that the Japanese are not there they are really and truly
not there. this becomes
particularly acute of day 1 is the rainy day. Can
reconnaissance be 100% effective at the same time as bombing is 100%
ineffective ?
But anyways K
enney made a decision by brute logic without the aid of game theory. The basis of his decision is best illustrated by the typical matrix used in game theory, outlined below(please excuse the limitations of the blogger format):
Japanese-------------------------------------------------------------------North South-------------------------------------------------------------------Allies North 2 2South 1 3-------------------------------------------------------------------The game is "zero sum". Whatever the Allies gain the Japanese lose. The above matrix represents the days of bombing available given the choices of both the Japanese to go either north or south and the choice of
Kenney to sent his
reconnaissance planes either north or south. From the allied point of view sending the recon planes north would give two days of bombing in either case, and sending them south would give either one or three days depending upon what the Japanese did. Hard to choose from these possibilities. From the Japanese point of view, however, the choice is obvious. Going south results in a loss of either -2 or -3 (the Japanese "gains" are the obverse of the allied gains tabled above). Going north results in a loss of either -2 or -1. The best strategy is to go north
no matter what the allies do. This indeed what happened in the real world. Both the convoy and the recon planes went north.
The secret of Von
Neumann's proof is that it applied only to those cases where "
the opponent plays as well as possible". It doesn't apply to an erratic opponent. Because of this there is a "
minimax" strategy for the allies. Send the planes to the north because the Japanese are not fools. And maybe they aren't good at "bluff" either. Both "bluff" and "stupidity" are complications that often occur in real life.
Where Von
Neumann's concept of "
minimax" really becomes interesting is where the game is not, like the example cited above, a "one shot deal" but is repeated.
Siegried uses an example modified from one given in Martin D Davis' book 'Game Theory:A Non-Technical Introduction' (Dover,
Mineola NY 1983/1997). It goes as follows:
Two players, call them Alice and Bob are each dealt a single card. Black always beats red. Each player antes up "5" on each round so there is always "10" in the pot. Alice plays first, and she can either call or bet an additional "3". If she calls both players show their cards, and the black card wins the pot. If they both have black or they both have red they split the pot 50/50. Bob, on the other hand can, if Alice bets, either fold or match her "3". If he folds Alice takes the "13" in the pot. If he matches and calls the one with the black card wins the pot, and if both players have the same colour they once more split the pot 50/50.
The matrix for this game is considerably more complex than that given above. I'll spare you the attempt, but try it out if you are interested. Instead of the 4 possible entries in the matrix above there are 128 different possibilities because both Alice and Bob can follow 4 different strategies. From Alice's point of view she can either 1)always bet, 2)always call, 3)bet with red and pass with black or 4)bet with black and pass with red. The result with Bob's options is a 4X4 matrix with 8 entries in each cell because the different winnings have to be listed beside each other. This sort of game introduces the idea of "bluff", and it turns out that the best
minimax strategy is a "mixed" one for both players. Alice's best strategy is to bet 60% of the time no matter what card she has (
ie to "bluff" if she has black) and 40% of the time to bet only if she has black. Bob, on the other hand, should call Alice's bet 40% of the time no matter what card he has and 60% of the time call if he has black and fold if he has red. The choice of what to do should be totally random so that the percentages equal the above.
This game, by the way, is stacked in favour of Alice. She'll come away with a gain of about 30% if both players play their optimum strategies. So Von
Neumann showed that there are
minimax strategies that are "mixed strategies" where a given course of action is chosen randomly a certain percentage of the time.
Molly
