Game Theory: An Introductory Sketch

The Paradox of Benevolent Authority

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The "Prisoners' Dilemma" is without doubt the most influential single analysis in Game Theory, and many social scientists, philosophers and mathematicians have used it as a justification for interventions by governments and other authorities to limit individual choice. After all, in the Prisoners' Dilemma, rational self-interested individual choice makes both parties worse off. A difficulty with this sort of reasoning is that it treats the authority as a deus ex machina -- a sort of predictable, benevolent robot who steps in and makes everything right. But a few game theorists and some economists (influenced by Game Theory but not strictly working in the Game Theoretic framework) have pointed out that the authority is a player in the game, and that makes a difference. This essay will follow that line of thought in an explicitly Game-Theoretic (but very simple) frame, beginning with the Prisoners' Dilemma. Since we begin with a Prisoners' Dilemma, we have two participants, whom we will call "commoners," who interact in a Prisoners' Dilemma with payoffs as follows:

Table 16-1

The third player in this game is the "authority," and she (or he) is a very strange sort of player. She can change the payoffs to the commoners. The authority has two strategies, "penalize" or "don't penalize." If she chooses "penalize," the payoffs to the two commoners are reduced by 7. If she chooses "don't penalize," there is no change in the payoffs to the two commoners.

The authority also has two other peculiar characteristics:

• She is benevolent: the payoff to the authority is the sum of the payoffs to the commoners.
• She is flexible: the authority chooses her strategy only after the commoners have chosen theirs.

Now suppose that the authority chooses the strategy "penalize" if, and only if, one or both of the commoners chooses the strategy "defect." The payoffs to the commoners would then be

Table 16-2

But the difficulty is that this does not allow for the authority's flexibility and benevolence. Is that indeed the strategy the authority will choose? The strategy choices are shown as a tree in Figure 1 below. In the diagram, we assume that commoner 1 chooses first and commoner 2 second. In a Prisoners' Dilemma, it doesn't matter which participant chooses first, or they both choose at the same time. What is important is that the authority chooses last.

Figure 16-1

What we see in the figure is that the authority has a dominant strategy: not to penalize. No matter what the two commoners choose, imposing a penalty will make them worse off, and since the authority is benevolent -- she "feels their pain," her payoffs being the sum total of theirs -- she will always have an incentive to let them off, not to penalize. But the result is that she cannot change the Prisoners Dilemma. Both commoners will choose "defect," the payoffs will be (5,5) for the commoners, and 10 for the authority.

Perhaps the authority will announce that she intends to punish the commoners if they choose "defect." But they will not be fooled, because they know that, whatever they do, punishment will reduce the payoff to the authority herself, and that she will not choose a strategy that reduces her payoffs. Her announcements that she intends to punish will not be credible.

EXERCISE In this example, a punishment must fall on both commoners, even if only one defects. Does this make a difference for the result? Assume instead that the authority can impose a penalty on one and not the other, so that the authority has 4 strategies: no penalty, penalize commoner 1, penalize commoner 2, penalize both. What are the payoffs to the authority in the sixteen possible outcomes that we now have? Under what circumstances will a benevolent authority penalize? What are the equilibrium outcomes in this more complicated game?

There are two ways to solve this problem. First, the authority might not be benevolent. Second, the authority might not be flexible.

Non-benevolent authority:

We might change the payoffs to the authority so that the authority no longer "feels the pain" of the commoners. For example, make the payoff to the authority 1 if both commoners cooperate and zero otherwise. We might call an authority with a payoff system like this a "Prussian" authority, since she values "order" regardless of the consequences for the people, an attitude sometimes associated with the Prussian state. She then has nothing to lose by penalizing the commoners whenever there is defection, and announcements that she will penalize defection become credible. EXERCISE Suppose the authority is sadistic; that is, the authority's payoff is 1 if a penalty is imposed and 0 otherwise. What will be the game equilibrium in this case?

Non-flexible authority:

If the authority can somehow commit herself to imposing the penalty in some cases and not in others, perhaps by posting a bond greater than the 15 point cost of a penalty, then the announcement of an intention to penalize would become credible. The announcement and commitment would then be a strategy choice that the authority would make first, rather than last. Let's say that at the first step, the authority has two strategies: commit to a penalty whenever any commoner chooses "defect," or don't commit. We then have a tree diagram like Figure 2. What we see in Figure 2 is that if the authority commits, the outcome will be cooperation and a payoff of 20 for her, at the top; but if she does not commit, the outcome will be at the bottom -- both commoners defect and the payoff will be -4 for the authority. So the authority will choose the strategy of commitment, if she can, and in that case the rational, self-interested action of the commoners will lead to cooperation and good results. But, if the commoners irrationally defect, or if they don't believe the commitment and defect for that reason, then the authority is boxed in. She has to impose a penalty even though it makes everyone worse off. In short, she cannot be flexible.

Figure 16-2

What we have seen here are two principles that play an important part in modern macroeconomics. Many modern economists apply these principles to the central banks that control the money supply in modern economies. They are

The principle of "rules rather than discretion."

That is, the authority should act according to rules chosen in advance, rather than responding flexibly to events as they occur. In the case of the central banks, they should control the money supply or the interest rate on public debt (there is controversy about which) according to some simple rule, such as increasing the money supply at a steady rate or raising the interest rate when production is close to capacity, to prevent inflation. If some groups in the economy push their prices up, the monetary authority might be tempted to print money, which would cause inflation and help other groups to catch up with their prices, and perhaps reduce unemployment. But this must be avoided, since the groups will come to anticipate it and just push their prices up all the faster.

The principle of credibility.

It is not enough for the authority to be committed to the simple rule. The commitment must be credible if the rule is to have its best effect.

The difficulty is that it may be difficult for the authority to commit itself and to make the commitment credible. This can be illustrated by another application: dealing with terrorism. Some governments have taken the position that they will not negotiate with terrorists who take hostages, but when the terrorists actually have hostages, the pressure to make some sort of a deal can be very strong. What is to prevent a sensitive government from caving in -- just this once, of course! And potential terrorists know those pressures exist, so that the commitments of governments may not be credible to them, even when the governments have a "track record" of being tough.

This may have an effect on the way we want our institutions to function, at the most basic, more or less constitutional level. For example, in countries with strong currencies, like Germany and the United States, the central bank or monetary authority is strongly insulated from democratic politics. This means that the pressures for a more "flexible" policy expressed by voters are not transmitted to the monetary authority -- or, anyway, they are not as strong as they might otherwise be -- so the monetary authority is more likely to commit itself to a simple rule and the commitment will be more credible.

Are these "conservative" or "liberal" ideas? Some would say that they are conservative rather than liberal, on the grounds that liberals believe in flexibility -- considering each case on its own merits, and making the best decision in the circumstances, regardless of unthinking rules. But it may be a little more complex than that. This and the previous essay have considered particular cases in which commitment and rules work better than flexibility. There may be many other cases in which flexibility is needed. I should think that the "liberal" approach would be to consider the case for commitment and for rules rather than discretion on its merits in each instance, rather than relying on an unthinking rule against rules! Anyway, conservative or liberal or radical (as it could be!), the theory of games in extended form is now a key tool for understanding the role of commitment and rules in any society.

Roger A. McCain