Game Theory using Nash Equilibrium and Maxmin/Minmax Strategy in History
In lecture we talked about Game Theory and we can see many manifestations of this theory in real life. One particular application of game theory is in the area of armed conflict/war, where it is widely believed to be a zero-sum game. A zero sum game is typically defined when each player’s gain or loss is exactly balanced by the gain or loss of other player’s. After each player’s sum up their gains and losses, the eventual result is zero. The zero-sum game is often solved with Minmax theorem or with Nash Equilibrium.
Let’s use the famous battle of Mismark Sea that occurred in March 1943 as the background to illustrate this point. The story was that during WWII, the Japanese wanted to reinforce New Guinea by sending more troops to the west. To reach there, they must decide if they want to go by the North or South Coast of New Britain Island. The U.S (the second player) has aircraft on the West of the New Britain Island and had to decide if it were to focus reconnaissance on the North or Southern Side of the island. The North side of the island is under heavy cloud, hence is able to provide good cover while the Southern side is clear. Of course, once the U.S locates the Japanese, they would launch their attacks. Hence in this case, the U.S want to minimize, the Japanese wants to maximize the amount of time that the Japanese ships are exposed to attacks.
In this case, choosing the cloud-covered Northern route may seem to be a strategy for the Japanese to maximize their payoff. However, assuming the U.S is also aware of the strategy, the decisive factor of the game then predicates on the thickness of the cloud cover. We should first introduce a payoff matrix to the game.
Japan
US
| North | South | |
| North | 2 | 3 |
| South | 1 | 4 |
If the payoff matrix above represents the number of days of bombing of the Japanese ships by U.S, then the U.S wants to maximize it, and the Japanese wants to minimize it. The dominant strategy for Japanese is obviously to sail North, and the U.S, by similar reasoning, also chooses north, which essentially creates a Nash Equilibrium of 2 days of U.S bombing.
Interestingly, there is actually another way to think about this game using the Maxmin/Minmax approach, which is based on the premise that in a zero-sum game, the other player is trying to maximize his own payoff through minimizing your payoff. Hence, a zero-sum game can be played with the goal of maximizing payoff for the row player and to minimize payoff for the column player by the row player. The row players looks at the worst case scenario for each of his possible strategies and chooses the strategy that offers the maximum of the minimum payoffs, such that he wins at least the maximum payoff he can attain, regardless of the column player’s strategies. Similarly, the column player looks at the worst that can happen when he plays his strategies and decides on the strategy that gives the minimum of the maximum payoffs. The U.S might think: What is the greatest number of bombing days we can guarantee, regardless of what the Japanese does? And the answer is 2, by choosing North.
By similar reasoning, the Japanese will reason that the lowest number of days guaranteed of bombing is 2, also by choosing North.
This happens so that the column player, the Japanese in this case, is able to keep row player, the U.S’s payoff down to his Minmax payoff, regardless of the U.S’s strategies. In this case, the Maxmin Payoff=Minmax Payoff, and this zero-sum game is strictly determined, in the sense that neither player in the game could outsmart the other opponent by determining his strategy, and neither player will benefit from deviating from his choice, since the outcome is always a Nash Equilibrium.
Article: http://personal.stthomas.edu/csmarcott/Ec355/kenny.pdf
http://en.wikipedia.org/wiki/Zero-sum_game
userpages.umbc.edu/~nmiller/POLI388/ZERO–SUM%20GAMES.pp