Game Theory and Networks

An Example of a Game-Theoretic Analysis

One of the main premises of game theory to predict behavior is that the behavior of all actors should be in an equilibrium. The Nash equilibrium is the best-known equilibrium concept. Actors' behavior satisfies Nash equilibrium conditions if, given the behavior of other actors, no actor can improve his situation by unilaterally changing his behavior (or strategy, in game-theoretic terminology). Many game-theoretic analyses are straightforward if one considers only two actors who interact with each other only once. One example is a trust game, as shown in Figure 1. Actor A has to decide first whether or not to trust Actor B. If A does not trust B, both actors get nothing. If A trusts B but B abuses trust, A incurs a loss, while B receives a large gain. If A trusts B and B honors trust, both A and B obtain a small gain. Because B is the last to decide, it is convenient to start the analysis with the decision of B, which in game-theoretic terms is called backward induction. If A trusts B, B has to decide between a large and a small gain. Given that A and B do not have any relation to each other any longer after this interaction, B should choose the large gain and, thus, abuse trust. Because A realizes this, he has to choose between incurring a loss and obtaining nothing. This implies that A prefers obtaining nothing, which means that he does not trust B. In Figure 1, the bold lines indicate the predicted moves for each of the [Page 304]decisions to be made in the trust game. As a result, A and B both get nothing, while they both could have had a small gain if A had trusted B and B would have honored trust. In this sense, the trust game is an example of a social dilemma.

Effects of Networks

Game-theoretic analyses become much more complicated if more actors are involved and if actors interact repeatedly with each other. Both features are typically present in networks. If A and B in the trust game interact repeatedly, there may exist equilibria of the type that A trusts B as long as B honors trust, but A never trusts B after any abuse of trust by B. Conditions on repeated game equilibria can be found in most textbooks on game theory. The theoretical arguments of repeated games can be extended to networks, such as in the following example: there are many different Actors A who all have repeated interactions with the same Actor B. Between interactions, Actors A can communicate about their experiences with B if they have a relation. Often, the crucial mechanism behind network effects is the diffusion of information via links in the network. In this case, whether B has incentives to honor the trust of a particular A depends on the network position of this Actor A and B's expectation about how many other Actors A will be informed if he abuses the trust of this A. Using these arguments about trust and networks, new equilibria can be derived that lead to predictions about in which networks and in which particular network positions Actors A can trust B more or less easily, as shown by Vincent Buskens. For example, denser networks provide better opportunities for trust, and actors in more central network positions can trust others more easily than actors in more peripheral network positions.

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