Saturday, April 28, 2007

Game theory: Nash Equilibrium

We have applied game theory from a simpler Prisoners' Dilemma situation to a more realistic game model in a real-world example of strategic thinking, say choosing an information system in the followings.

For this example, the players will be a company considering the choice of a new information system, and a supplier who is considering producing it. The two choices are to install a technically advanced or a more proven system with less functionality. We'll assume that the more advanced system really does supply a lot more functionality.

Again, in this case, we can express all this compactly in a payoff table. Basically, the table would indicate that if both the company and supplier choose the technically advanced information system, each earns $20 million in profits from the system, but if the company chooses the advanced system and the supplier does not choose to produce it or vice versa, then both earn nil profits for the period under consideration. However, if both choose the proven information system, then both earn only $5 million of profits each.

We see that both players can be better off, on net, if an advanced system is installed. But the worst that can happen is for one player to commit to an advance system while the other player stays with the proven one. In that case there is no deal, and no payoffs for anyone. The problem is that the supplier and the user must establish a compatible standard, in order to work together, and since the choice of a standard is a strategic choice, their strategies have to mesh.

Although it looks a lot like the Prisoners' Dilemma at first glance, this is a more complicated game. We'll take several complications in turn:

1. By observing the table carefully, we would notice that there are no dominated strategies in this game. The best strategy for each participant depends on the strategy chosen by the other participant. Thus, we need a new concept of game-equilibrium that will allow for that complication. When there are no dominant strategies, we often use an equilibrium conception called the Nash Equilibrium, named after Nobel Memorial Laureate John Nash.

2. Nash Equilibrium occurs when there is a set of strategies with the property that no player can benefit by changing his / her strategy while the other players keep their strategies unchanged. In this case, this set of strategies and the corresponding payoffs constitute the Nash Equilibrium.

3. The Nash Equilibrium is a pretty simple idea: we have a Nash Equilibrium if each participant chooses the best strategy, given the strategy chosen by the other participant. In the example, if the user opts for the advanced system, then it is best for the supplier to do that too. So (Advanced, Advanced) is a Nash-equilibrium.

4. If the user chooses the proven system, it's best for the supplier to do that too. There are as such two Nash Equilibria. It may seem easy enough to opt for the advanced system which is better all around, but if each participant believes that the other will stick with the proven system, then it will be best for each player to choose the proven system. This is a danger typical of a class of games called coordination games -- and what we have learned is that the choice of compatible standards is a coordination game.

5. We have assumed that the payoffs are known and certain. In the real world, every strategic decision is risky -- and a decision for the advanced system is likely to be riskier than a decision for the proven system. Thus, we would have to take into account the players' subjective attitudes toward risk, in other words their risk aversion, to make the example fully realistic.

6. The example assumes that payoffs are measured in money. Thus, we are not only leaving risk aversion out of the picture, but also any other subjective rewards and penalties that cannot be measured in money. Economists have ways of measuring subjective rewards in money terms. To simplify the analysis, we assume that all rewards and penalties are measured in money and are transferable from the user to the supplier and vice versa.

7. Real choices of information systems are likely to involve more than two players, at least in the long run. The user may choose among several suppliers, and suppliers may have many customers. That makes the coordination problem harder to solve. Suppose, for example, that "beta" is the advanced system and "VHS" is the proven system, and suppose that about 90% of the market uses "VHS." Then "VHS" may take over the market from "beta" even though "beta" is the better system. Many economists, game theorists and others believe this is a main reason why certain technical standards gain dominance.

8. On the other hand, the user and the supplier don't have to just sit back and wait to see what the other person does. They can sit down and talk it out, and commit themselves to a contract. In fact, they have to do so, because the amount of payment from the user to the supplier also has to be agreed upon. In other words, unlike the Prisoners' Dilemma, this is a cooperative game, not a non-cooperative game. On the one hand, that will make the problem of coordinating standards easier, at least in the short run. On the other hand, Cooperative games call for a different approach to solution.

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