Game Theory Part 2: Zero Sum Games

Zero Sum Games The following matrix defines a zero-sum game. Notice the sum of the payoffs to each player, at every outcome, adds to zero. AB X(2,-2) Y(1,-1)(3,-3) Player 2 (Column) Player 1 ( row ) If both players know all the possible strategies and the resulting payoffs and play their best possible strategy in terms of their own best interest, what will happen?

Zero Sum Games Each player can reason as follows: Player 2 could do better and will do no worse by selecting strategy A as compared to B. This is because the payoffs to player 2 (while still negative) are sometimes better and never worse for player 2 in column A compared to column B. Expecting this, player 1 would do better with strategy X. AB X(2,-2) Y(1,-1)(3,-3) Player 2 (Column) Player 1 ( row ) We are assuming each player expects the other to play the strategy in their own best interest. Each player asks: What is the best I can do assuming the other player does the best he can do?

Zero Sum Games AB X(2,-2) Y(1,-1)(3,-3) Player 2 (Column) Player 1 ( row ) We say that the strategies X and A form an equilibrium point because neither player can achieve a better result assuming the other player doesn’t change strategy. This combination of strategies is also called a saddle point because it is an equilibrium point where both players are getting the highest possible payoff assuming the best play by the opponent. Of course we don’t know what actual players in this game might do. But assuming each is trying to achieve the highest payoff, there is no reason to expect any other outcome.

Battle of the Bismark Sea The previous game was actually played out in February 1943 as the Battle of the Bismark Sea between U.S. and Japanese forces. We may consider the U.S. commander, General Kenny as player 1, while his opponent, player 2, is Admiral Imamura.

Battle of the Bismark Sea In the middle of World War II, the Japanese Admiral was ordered to deliver reinforcements to Japanese soldiers fighting in Papua New Guinea. The Japanese had to choose between two available routes…

Battle of the Bismark Sea The Japanese reinforcements could be sent either by a northern route, through the Bismark Sea, or through a southern route, through the Solomon sea. General Kenney expected the reinforcements to be sent and knew the supply routes available to the Japanese. Both commanders also knew of the significance of certain weather conditions. If the U.S. commander anticipated his opponent’s move, and sent his planes toward that route, he would have more days of bombing available.

Battle of the Bismark Sea If the U.S. commander guessed his opponent’s decision incorrectly, he would have to redirect his planes and would lose a day of bombing. Let’s say the “payoffs” in this game are the number of days of bombing: positive for the U.S. forces and negative for the Japanese. The number of days is the same for each, just positive for one and negative for the other. Hence, this is a zero-sum game. A deciding factor was the bad weather in the north, limiting the number of days the Americans could bomb the Japanese to only two days in the northern route.

The Battle of the Bismark Sea Travel North Travel South Search North (2,-2) Search South (1,-1)(3,-3) Admiral Imamura General Kenney We can put this game into matrix form as follows. The numbers at each outcome represent number of days of bombing, positive for the Americans and negative for the Japanese. As the name of the battle suggests, both commanders choose the northern route, through the Bismark Sea. In spite of this being an “optimal strategy” for the Japanese, they suffered heavy losses. However, the loss certainly would have been greater for the Japanese with another day of bombing.

Zero Sum Games Travel North Travel South Search North (2,-2) Search South (1,-1)(3,-3) Admiral Imamura General Kenney Normally, if a matrix game is a zero-sum game, we don’t need to write the payoffs for both players. Instead we write only one number in each matrix cell. If only one number is written in each cell of the matrix, then the game is understood to be a zero-sum game. Travel North Travel South Search North 22 Search South 13 Admiral Imamura General Kenney

Zero Sum Games Travel North Travel South Search North 22 Search South 13 Admiral Imamura General Kenney For example, in the matrix below, because only one number is written in each cell, it is understood that the payoffs are the given numbers for the row player (General Kenney) and the negative of each of these values for the column player (Admiral Imamura).

Finding Saddle Points Travel North Travel South Search North 22 Search South 13 Column Player Row Player For zero-sum games, we can look for saddle points by recognizing the following two facts: The row player will want to the maximum value in the matrix. The column player will want the minimum value in the matrix. Note that in zero sum games saddle points are the same as equilibrium points. A saddle point is where both players are getting the best result possible assuming the opponent is doing their best. An equilibrium point is where neither player has a reason to change strategies assuming the other player doesn’t change.

Travel North Travel South Search North 22 Search South 13 Column Player Row Player To look for a saddle point in a matrix game, begin by finding the minimum values of each row. ( These are called the row minima. ) Then, choose the maximum value of these minima. (This is called the maximin strategy for the row player. ) We do this because a rationalization for the row player is to find the best move possible assuming that the column player chooses his best strategy. It’s like thinking “what’s the best I can do when he plays his best.” 2 1 Finding Saddle Points

Travel North Travel South Search North 22 Search South 13 Column Player Row Player Now, look for the maximum values for each column. (These are column maxima.) Then, choose the smallest of the column maxima. This is called the minimax strategy for the column player. Here, the column player is thinking: “What is the best I can do when the row player is playing his best strategy.” Finding Saddle Points

Travel North Travel South Search North 22 Search South 13 Column Player Row Player Because, in this example, the row maximin and the column minimax strategies coincide with the same outcome, these strategies are called a saddle point (or an equilibrium point). In some cases, given a 2x2 matrix game, we can find equilibria by determining if the row maximin and column minimax coincide with one outcome Finding Saddle Points

Travel North Travel South Search North 22 Search South 13 Column Player Row Player The “equilibrium point” or “saddle point” for this game is the combination of the strategies of traveling and searching north. It is the best each player can do assuming the other player does their best. The outcome associated with the equilibrium point is called the value of the game. In this case, the value is 2. We say a game is fair if it’s value is 0. Thus, this particular game is not fair. equilibrium point Finding Saddle Points

Saddle Points In zero-sum games, the terms saddle point and equilibrium point are interchangeable. These terms refer to the combination of strategies that are in each player’s best interest assuming all other player’s use the strategy in their best interest. The term saddle point comes from the fact that, in a game with two players, each with two strategies, it represents at one point a highest point for the lowest payoff values and also a lowest point for the highest payoff values. Row player wants highest value in matrix Column player wants lowest value in matrix

Saddle Points As the row player, imagine these lines are your choices. You want the highest on the curve. As the column player, imagine these lines as your choices. You want the lowest on the curve. The saddle point is where both players get the best result assuming the other makes the best choice for themselves.

Saddle Points As the row player, imagine these lines are your choices. You want the highest on the curve. As the column player, imagine these lines as your choices. You want the lowest on the curve. The column player is picking the line that is a low as possible and the row player is picking the line that is as high as possible.

Saddle Points As the row player, imagine these lines are your choices. You want the highest on the curve. As the column player, imagine these lines as your choices. You want the lowest on the curve. The saddle point is an equilibrium point because neither player would have a reason to make a different choice assuming the other player didn’t make a different choice.