by the columns, and each row/column represents a possible strategy. The cells of the matrix contain the payoffs for both players. (The first number represents the player in the rows, the second number the player in the columns.)
| Player B | |||
| Strategy B1 | Strategy B2 | ||
| Player A | Strategy A1 | 10, 2 | 9, 10 |
| Strategy A2 | 12, 3 | 11, 0 |
For example, in the game depicted above, there are two players, Player A and Player B. Both have two strategies, denoted A1, A2 for A, and B1, B2 for B. The inside of the matrix contains the payoffs; for example, if Player A plays A1 and Player B plays B1, the payoffs are 10 and 2, so Player A gets 10, and Player B gets 2.
If there is perfect information, both players are aware of all of the payout matrix possibilities and strategies available to each player. In a situation like this, the game is solved by analyzing each player’s potential actions.
Player A’s payouts depend on two things, his own choices and the choices of Player B. While Player A does not know what Player B wants, he can determine what the best choices for himself would be if Player B chooses B1 or B2. If Player B plays B1, then Player A could choose A1 for 10 but selecting A2 would yield him 12, which is better:
| Player B | |||
| Strategy B1 | Strategy B2 | ||
| Player A | Strategy A1 | 10, 2 | 9, 10 |
| Strategy A2 | 12, 3 | 11, 0 |
Similarly, if Player B plays B2, choosing A1 would net Player A 9, but choosing A2 would yield 11, which is better:
| Player B | |||
| Strategy B1 | Strategy B2 | ||
| Player A | Strategy A1 | 10, 2 | 9, 10 |
| Strategy A2 | 12, 3 | 11, 0 |
In this specific game, Player A has a dominant strategy: Playing A2 is better than playing A1 no matter what:
| Player B | |||
| Strategy B1 | Strategy B2 | ||
| Player A | Strategy A1 | 10, 2 | 9, 10 |
| Strategy A2 | 12, 3 | 11, 0 |
Player B’s thinking is the same as Player A’s, so he selects the best strategy for every one of Player A’s potential strategies. If Player A plays A1, Player B must choose between 2 and 10; B2 is better. If Player A chooses A2, Player B chooses between 3 and 0, and B1 is better:
| Player B | |||
| Strategy B1 | Strategy B2 | ||
| Player A | Strategy A1 | 10, 2 | 9, 10 |
| Strategy A2 | 12, 3 | 11, 0 |
As opposed to Player A, Player B has no dominant strategy. B1 is better if Player A plays A2; B2 is better if Player A plays A1.
However, this is perfect information. Player B not only knows his own payoffs, he knows Player A’s payoffs as well. Because of this, Player B also knows that Player A has a dominant strategy, A2. This knowledge enables Player B to select the best outcome possible for himself, B1:
| Player B | |||
| Strategy B1 | Strategy B2 | ||
| Player A | Strategy A1 | 10, 2 | 9, 10 |
| Strategy A2 | 12, 3 | 11, 0 |
In the above game, the A2, B1 combination is what is called a Nash-equilibrium. In a Nash-equilibrium, neither player can unilaterally improve their own position. In the above example, if Player A plays A1 instead of A2, he loses 2; if Player B plays B2 instead of B1, he loses 3.
Note that this is not the socially optimal outcome. The combined gains for the two participants is 15 while playing A2B1, but it would be 19 while playing A1B2, which would be a 27% increase. However, that point is not reachable, because for Player A, it is better to stick to A2. If, however, some kind of mediation is available for Player B, the game could be changed for a greater combined outcome beneficial to both players.
Case Problem
In the game above, a mediation service is available. Both participants are aware of the service and trust the service provider. Assume further that the mediation costs 1. How much will Player B end up with?
Sequential Games
In sequential games (or extensive form games), the players take turns acting. This means that they are not only aware of the strategies and the payoffs but also the prior actions of all players.
Extensive form games can be depicted by a tree graph. Each node (or vertex) represents a choice for a player. The player is depicted on each node, the choices are on the edges leading out of the nodes, the payoffs are written on the “leaves,” at the end of each tree branch.
In the example above, Player 1 moves first, potentially choosing between 1a and 1b. If Player 1 chooses 1a, he goes to the upper node 2, giving Player 2 the options 2a or 2b. Had Player 1 chosen 1b, then Player 2 could choose between 2c and 2d.
Solving an extensive form game starts from the outcome leaves. Player 1 knows that Player 2 will choose the outcome that is more beneficial for him, so if Player 2 has to choose between 2a (a payout of 2) and 2b (a payout of 3), he will choose 2b. Similarly, if Player 2 has to choose between 2c and 2d, he will choose 2d. At that point, all Player 1 has to do is to compare their payoffs at the leaf nodes of 2b (a payout of 3) and 2d (a payout of 2) to find the best option. He will play 1a, 2b, with payouts of 3 and 3. As long as there is no mechanism for the two players to trust each other, they will keep playing such (self-serving) strategies. This outcome is by far the worst for their society; the sum of the payouts is 6, while the other options offer 12, 22, or 29. Again, it is clear that a mediation system could benefit both participants.
Moral