The theory of zero-sum games is vastly different from that of non-zero-sum games because an optimal solution can always be found. However, this hardly represents the conflicts faced in the everyday world. Problems in the real world do not usually have straightforward results. The branch of Game Theory that better represents the dynamics of the world we live in is called the theory of non-zero-sum games. Non-zero-sum games differ from zero-sum games in that there is no universally accepted solution. That is, there is no single optimal strategy that is preferable to all others, nor is there a predictable outcome. Non-zero-sum games are also non-strictly competitive, as opposed to the completely competitive zero-sum games, because such games generally have both competitive and cooperative elements. Players engaged in a non-zero sum conflict have some complementary interests and some interests that are completely opposed.
The Battle of the Sexes is a simple example of a typical non-zero-sum game. In this example a man and his wife want to go out for the evening. They have decided to go either to a ballet or to a boxing match. Both prefer to go together rather than going alone. While the man prefers to go to the boxing match, he would prefer to go with his wife to the ballet rather than go to the fight alone. Similarly, the wife would prefer to go to the ballet, but she too would rather go to the fight with her husband than go to the ballet alone. The matrix representing the game is given below:
| Husband | |||
|
Boxing Match |
Ballet |
||
| Wife |
Boxing Match |
2, 3 |
1, 1 |
|
Ballet |
1, 1 |
3, 2 |
|
The wife's payoff matrix is represented by the first element of the ordered pair while the husband's payoff matrix is represented by the second of the ordered pair.
From the matrix above, it can be seen that the situation represents a non-zero-sum, non-strictly competitive conflict. The common interest between the husband and wife is that they would both prefer to be together than to go to the events separately. However, the opposing interests is that the wife prefers to go to the ballet while her husband prefers to go to the boxing match.
It is conventional belief that the ability to communicate could never work to a player's disadvantage since a player can always refuse to exercise his right to communicate. However, refusing to communicate is different from an inability to communicate. The inability to communicate may work to a player's advantage in many cases.
An experiment performed by Luce and Raiffa compares what happends when player can communicate and when players cannot communicate. Luce and Raiffa devised the following game:
|
a |
b |
|
|
A |
1, 2 |
3, 1 |
|
B |
0, -200 |
2, -300 |
If Susan and Bob cannot communicate, then there is no possiblity of threats being made. So, Susan can do no better than to play strategy A and Bob can do no better than to play strategy a. Susan, therefore gains 1 and Bob gains 2. However, when communication is allowed, the situation is complicated. Susan can threaten Bob by saying that she will play strategy B unless Bob commits himself to playing strategy b. If Bob submits, Susan will gain 2 and Bob will lose 1 (as opposed to Susan gaining 1 and Bob gaining 2 when communication is not allowed).
The Battle of the Sexes example given above seems to be an unsolvable dilemma. However, this problem can be solved it either the husband or the wife resticts the others' alternatives. For example, if the wife buys two tickets for the ballet, indicating that she is definitely not going to the boxing match, the husband would have to go to the ballet along with his wife in order to maximize his self-interest. Because the wife bought the two tickets, the husbands optimal payoff, now, would be to go along with his wife. If he were to go to the boxing match alone, he would not be maximizing his self-interest.
If the game is played only once, players do not have to fear retaliation from their opponents, so they may play differently than they would in a game played repeatedly.
