WHAT IS ZERO-SUM? DICTATOR GAME? TRUST GAME?

Posted by: lizwang on: June 23, 2008

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In game theory, zero-sum describes a situation in which a participant’s gain or loss is exactly balanced by the losses or gains of the other participant(s). It is so named because when the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero. Chess and Go are examples of a zero-sum game: it is impossible for both players to win. Zero-sum can be thought of more generally as constant sum where the benefits and losses to all players sum to the same value. Cutting a cake is zero- or constant-sum because taking a larger piece reduces the amount of cake available for others. In contrast, non-zero-sum describes a situation in which the interacting parties’ aggregate gains and losses is either less than or more than zero.

Situations where participants can all gain or suffer together, such as a country with an excess of bananas trading with another country for their excess of apples, where both benefit from the transaction, are referred to as non-zero-sum. Other non-zero-sum games are games in which the sum of gains and losses by the players are always more or less than what they began with. For example, a game of poker, disregarding the house’s rake, played in a casino is a zero-sum game unless the pleasure of gambling or the cost of operating a casino is taken into account, making it a non-zero-sum game.

The concept was first developed in game theory and consequently zero-sum situations are often called zero-sum games though this does not imply that the concept, or game theory itself, applies only to what are commonly referred to as games. In pure strategies, each outcome is Pareto optimal (generally, any game where all strategies are Pareto optimal is called a conflict game) [1]. Nash equilibria of two-player zero-sum games are exactly pairs of minimax strategies.

In 1944 John von Neumann and Oskar Morgenstern proved that any zero-sum game involving n players is in fact a generalized form of a zero-sum game for two players, and that any non-zero-sum game for n players can be reduced to a zero-sum game for n + 1 players; the (n + 1) player representing the global profit or loss. This suggests that the zero-sum game for two players forms the essential core of mathematical game theory.

Many economic situations are not zero-sum, since valuable goods and services can be created, destroyed, or badly allocated, and any of these will create a net gain or loss. Assuming the counterparties are acting rationally, any commercial exchange is a non-zero-sum activity, because each party must consider the goods s/he is receiving as being at least fractionally more valuable to him/her than the goods he/she is delivering. Economic exchanges must benefit both parties enough above the zero-sum such that each party can overcome his or her transaction costs.

The most common or simple example from the subfield of Social Psychology is the concept of "Social Traps". In some cases we can enhance our collective well-being by pursuing our personal interests — or parties can pursue mutually destructive behavior as they choose their own ends.

It has been theorized by Robert Wright, among others, that society becomes increasingly non-zero-sum as it becomes more complex, specialized, and interdependent. As former US President Bill Clinton states:

The more complex societies get and the more complex the networks of interdependence within and beyond community and national borders get, the more people are forced in their own interests to find non-zero-sum solutions. That is, win–win solutions instead of win–lose solutions…. Because we find as our interdependence increases that, on the whole, we do better when other people do better as well — so we have to find ways that we can all win, we have to accommodate each other…. Bill Clinton, Wired interview, December 2000.

	A	B	C
1	30, -30	-10, 10	20, -20
2	10, -10	20, -20	-20, 20

A game’s payoff matrix is a convenient representation. Consider for example the two-player zero-sum game pictured at right.

The order of play proceeds as follows: The first player (red) chooses in secret one of the two actions 1 or 2; the second player (blue), unaware of the first player’s choice, chooses in secret one of the three actions A, B or C. Then, the choices are revealed and each player’s points total is affected according to the payoff for those choices.

Example: Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 20 points and Blue loses 20 points.

Now, in this example game both players know the payoff matrix and attempt to maximize the number of their points. What should they do?

Red could reason as follows: "With action 2, I could lose up to 20 points and can win only 20, while with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, Blue would choose action C. If both players take these actions, Red will win 20 points. But what happens if Blue anticipates Red’s reasoning and choice of action 1, and deviously goes for action B, so as to win 10 points? Or if Red in turn anticipates this devious trick and goes for action 2, so as to win 20 points after all?

John von Neumann had the fundamental and surprising insight that probability provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. Each player computes the probabilities so as to minimise the maximum expected point-loss independent of the opponent’s strategy. This leads to a linear programming problem with a unique solution for each player. This minimax method can compute provably optimal strategies for all two-player zero-sum games.

For the example given above, it turns out that Red should choose action 1 with probability 57% and action 2 with 43%, while Blue should assign the probabilities 0%, 57% and 43% to the three actions A, B and C. Red will then win 2.85 points on average per game.

 

The dictator game is a very simple game in experimental economics, similar to the ultimatum game. Experimental results in the dictator game have often been cited as a conclusive rebuttal of the rationally self-interested individual (homo economicus) model of economic behavior, although this conclusion is controversial.

In the dictator game, the first player, "the proposer," determines an allocation (split) of some endowment (such as a cash prize). The second player, the "responder," simply receives the remainder of the endowment not allocated by the proposer to himself. The responder’s role is entirely passive (he has no strategic input into the outcome of the game). As a result, the dictator game is not formally a game at all (as the term is used in game theory). To be a game, every players’ outcome must depend on the actions of at least some others. Since the proposer’s outcome depends only on his own actions, this situation is one of decision theory and not game theory. Despite this formal point, the dictator game is used in the game theory literature as a degenerate game.

This "game" has been used to test the homo economicus model of individual behavior: if individuals were only concerned with their own economic well being, proposers (acting as dictators) would allocate the entire good to themselves and give nothing to the responder. Experimental results have indicated that individuals often allocate money to the responders, reducing the amount of money they receive. These results appear robust, Henrich, et al. discovered in a wide cross cultural study that proposers do allocate a non-zero share of the endowment to the responder.

If these experiments appropriately reflect individuals’ preferences outside of the laboratory, these results appear to demonstrate that either:

Proposers fail to maximize their own expected utility, or Proposer’s utility functions include benefits received by others.
Additional experiments have shown that subjects maintain a high degree of consistency across multiple versions of the dictator game in which the cost of giving varies. This suggests that dictator game behavior is, in fact, altruism instead of the failure of optimizing behavior. Other experiments have shown a relationship between political participation and dictator game giving, suggesting that it may be an externally valid indicator of concern for the well-being of others.

Some authors have suggested that giving in the dictator game does not entail that individuals wish to maximize other’s benefit (altruism). Instead they suggest that individuals have some negative utility associated with being seen as greedy, and are avoiding this judgment by the experimenter. Some experiments have been performed to test this hypothesis with mixed results.

Further experiments testing experimental effects have been performed. Bardsley has performed experiments where individuals are given the opportunity to give money, give nothing, or take money from the respondent. In these cases individuals consistently give less (close to zero). Bardsley suggests two interpretations for these results. First, it may be that the range of options cues experimental subjects to use different reasoning patterns. "Subjects might perceive dictator games as being about giving, since they can either do nothing or give, and so ask themselves how much to give. Whilst the taking game… might appear to be about taking for analogous reasons, so subjects ask themselves how much to take." Second, subjects behavior may be affected by a kind of framing effect. What a subject considers to be an appropriately kind behavior depends on the range of behaviors available. In the taking game, the range includes worse alternatives than the dictator game, and as a result giving less appears equally kind.

The trust game extends the dictator game one step by having the reward that the dictator can (unilaterally) split between himself and a partner partially decided by an initial gift from that partner. The initial move is from the dictator’s partner, who must decide how much of her initial endowment to trust with him (in the hopes of receiving some of it back). Normally, she is encouraged to give something to the dictator through a specification in the game’s rules that her endowment will be increased by a factor from the researchers. The experiments rarely end in the subgame perfect Nash equilibrium of "no trust."