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The Two-Person, Non-Zero-Sum Game

In a two-person game, there are generally both competitive and cooperative elements: the interests of the players are opposed in some aspects and complementary in others.

Let's regard all two-person games as lying in a continuum,with the zero-sum games (totally competitive) at one extreme, and completely cooperative game at the other extreme.

The payoff matrix tells the whole story in zero-sum games. However, in non-zero-sum games, there are many "rules of the game" that markedly affect the character of the game, and these rules must be spelled out before you can talk about the game intelligently.

• Communication. Generally, the more cooperative the game - the more the players' interests coincide - the more significant is the ability to communicate. However, tacit communication (one player unilaterally adopts an apparently inferior strategy to invite another to cooperate) is generally ineffective because it is generally misunderstood or used. Also it is possible to communicate threats.
• The order of play. The game may be far from trivial even when one player learns the opponent's strategy. What's more, the advantage of having this information may turn out to be a disadvantage.
• The effect of imperfect information. It is often to a player's advantage to see that the partner is well informed (including both real or fake messages). Players can gain if they can convince their opponent that they have certain attitudes or capabilities, whether they really have them or not.
  
• The effect of restricting alternatives. Players are sometimes prevented from using some of their strategies. It is one of the paradoxes of non-zero-sum games that this restriction of a player's choice may be turned to that player's advantage. One can strengthen his position by limiting his alternatives.
  
• Threats. A threat is a statement that you will act in a certain way under certain conditions. Unlike restricted alternatives, this self-imposed restriction isn't binding; you can always change your minds. The purpose of a threat is to change someone's behavior. A threat is effective only to the extent that it is plausible. The greater the price the party making the threat must pay to carry it out, the less plausible the threat.
  
• Binding agreements and side payments. The possibility of making binding agreements (agreement made is enforced by the rules) has a strong influence on the character of the game. In some games it is possible for one player to affect the actions of another by offering a "side payment", a payment made "under the table".
  

The Prisoner's Dilemma

Each player has two basic choices: to act "cooperatively" or "uncooperatively". When all the players act cooperatively, each does better than when all of them act uncooperatively. For any fixed strategy (ies) of the other player(s), a player always does better by playing uncooperatively than by playing cooperatively.

If the game is played only once, players have no choice but to act uncooperatively. But when the game is played repeatedly infinitely, the cooperative strategy comes into practice.

For games repeated for a fixed number of trials, players behave the same as the once-off game. They will behave uncooperatively in the last trial, and thus inferring that the second last, and so on should not be played cooperatively.

The assumption of zero-sum game is that the opponent will act rationally. However, in non-zero-sum game, the assumption is that the partner is cooperative. The difference between the assumptions made in the zero-sum and in the non-zero-sum games is even more clear when they fail. In non-zero-sum games, cooperating with a partner who doesn't cooperative with you leads to disaster; in zero-sum games, the worst that can happen when you play minimax is that you lose an opportunity to swindle your opponent.

Utility Theory

A utility function is simply a "quantification" of a person's preferences with respect to certain objects.

Six conditions that guarantee the existence of a Utility Function

If a player's preferences are to be expressed by a utility function, these preferences must be consistent:

• Everything is comparable.
• Preference and indifference are transitive. That is, if A>B and B>C, then A>C.
• A player is indifferent when equivalent prizes are substituted in a lottery.
• A player will always gamble if the odds are good enough.
• The more likely the preferred prize, the better the lottery.
• Players are indifferent to gambling. A player's attitude toward a compound lottery - a lottery in which the prizes may be tickets to other lotteries - is dependent only on the ultimate prizes and the chances of getting them as determined by the laws of probability; the actual gambling mechanism is irrelevant.

It has even been suggested that these conditions should be used as a definition of rationality in decision making.

Restrictions

In reality, it is easy to show that a person cannot consistently prefer the one type of bet to be the other and still satisfy the six conditions necessary for consistency.

Another souce of difficulty is the variation in people's preferences over a period of time.

Experiments have shown that decisions often depend on seemingly irrelevant variables.

Context of a problem can influence decision making. It was found that most subjects made different decisions in two identical or equivalent situations when these situations were discribed in different ways. The general rule was this: pelple who feel they have won something generally try to conserve their winnings by avoiding risks. In an identical situation, the same people who perceive that they have just lost something will take risks they considered unacceptable before, to make themselves whole. Also people are more sensitive to the percentage of changes, but not the absolute changes.