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by Roger A. McCain
These notes are suggested by some recent discussions of the economic theory of persistent slumps, such as the Great Depression of the 1930's and the European unemployment of the 1980's and 1990's. Clearly, Keynesian ideas are a central (and still controversial) aspect of any such theory, but there are a wide variety of interpretation of Keynes, both favorable and critical, and recently one group in that broad tradition have begun to call themselves "Post-Walrasian." The Post-Walrasian view has several distinctive insights. The following list is not represented as either complete or final, but I hope it will do as a starting point:
These notes will attempt to give a game-theoretic framework for such a theory at about the level of simplicity of the introductory text. The discussion will be "simple" in that -- among other things -- it will not consider the multiple levels of analysis in the model itself, although aspects of the model will point in that direction.
Game theory seems promising for this purpose in part because the concept of Nash equilibrium is known to include a possibility of multiple equilibria. In my 1980 textbook I gave an illustration of games with multiple equilibria, some better than others, called the Heave-Ho Game. Here is the "little story" that goes with the game:
Two people are driving on a back-road short-cut and they come to a place where the road is obstructed by a fallen tree. Together, they are capable of moving the tree off the road, but only if each motorist heaves as hard as he can. If they do coordinate their decisions and heave, they can get the road clear, get back in their car, and continue to their destination. Let us say that their payoffs in that case are both +5. If neither motorist heaves, they have to turn back and arrive very late at their destination. In that case, we shall say that the payoffs are 0 for each motorist. However, if one motorist heaves and the other slacks off, making less than an all-out effort, the tree is shifted only a little -- it remains in the road -- the the motorist who heaved gets a painful muscle-strain as a result of the effort. Thus, the payoffs in that case are 0 (for the slacker) and -5 for the motorist who heaves. The payoff table is shown in Table 11-1.
Table 11-1
|
heave |
slack off |
|
|
heave |
5,5 |
-5,0 |
|
slack off |
0,-5 |
0,0 |
I hope it is clear that there are two Nash equilibria, at the upper left and the lower right. Starting from the upper right, either the column player or the row player will be worse off -- going from 5 to 0 -- if he changes strategies unilaterally. Starting from the lower right, again, either the column player or the row player will be worse off, going from 0 to -5, if he changes strategies unilaterally. However, the other two outcomes are not equilibria: from the upper right, for example, the row player will be better off to switch from a "heave" strategy, with a -5 payoff, to a "slack off" strategy, with a 0 payoff, if the other player does not change. Symmetrically, the lower left is not an equilibrium either.[1]
This reasoning -- that we have equilibrium if neither player can be better off by changing strategy unilaterally -- is consistent with the definition of Nash equilibrium. However, even though the lower right outcome is a Nash equilibrium, it is clearly inferior to the upper left outcome. In fact, it is Pareto-inferior, which means that a coordinated change of strategies from (slack off, slack off) to (heave, heave) would make some participants better off (in this case, both are better off) and nobody worse off. This illustrates that the game has multiple equilibria, with some equilibria superior to others. It also illustrates the importance of coordination: can they make the coordinated change of strategies they need to make them both better off? Because of this issue, the Heave-Ho game illustrates a pure coordination game.
With only two decision-makers to coordinate, it ought to be relatively easy. But even in this simple case, coordination might fail because of mistrust. If each of the motorists suspects that the other is a slacker, each will consider the choice as being between -5 and 0, rather than a choice between 0 and +5, and choose to slack off. Pessimism and loss aversion can have the same effect. A very pessimistic way of choosing strategies (in this situation) is to maximize the minimum payoff. "If I heave, my minimum payoff is -5. If I slack off, my minimum payoff is 0, and that is better than -5. I'll slack off."
We can also see how institutions might help to solve the coordination problem. An institution that might help in this case is a social convention. Suppose it were widely believe that, in cases of this kind, "gentlemen don't slack off. Just isn't done, don't y'know." If this convention were widespread enough that both motorists believed that the other motorist would subscribe to it -- and therefore would not slack off -- that would be enough to assure each motorist that the other one would slack off, and then each would behave like a gentleman -- and heave!
Because the Heave-Ho game is symmetrical, it doesn't quite tell the whole story. Let us make the example just a little more complex. Suppose that one of the motorists -- the "bridegroom" -- has more to lose by being late than the other motorist, the "hitchhiker." The payoff table for this modified game is Table 11-2.
Table 11-2
|
heave |
slack off |
|
|
heave |
5,5 |
-5,0 |
|
slack off |
0,-5 |
0,-4 |
In this case, the column player is the bridegroom and the row player is the hitchhiker. It is still true that they both gain in a coordinated switch of strategies from (slack off, slack off) to (heave, heave). But now the bridegroom gains 9, while the hitchhiker, who is in less of a hurry, gains only 5. Seeing that difference, the hitchhiker might demand some compensation for his cooperation -- turning the decision into a bargaining session. Bargaining outcomes can be unpredictable, and contribute to distrust, so this temptation could be another reason why coordination might fail.
Despite all this, it should be relatively easy for just two people to coordinate their strategy. In economics, we are often concerned with cases in which very large numbers of people must coordinate their strategies. We are also really concerned with macroeconomics, rather than back-road driving, and so it is time to bring the economics to the fore. Accordingly, we consider an investment game with N potential investors -- N very large -- and two strategies: each investor can choose a high rate of investment or a low one. We suppose that the payoff to a low rate of investment is always zero, but the payoff to a high rate of investment depends on how many other investors choose a high rate of investment.
The payoffs in the Investment Game are shown in Figure 9-1. The figure is an xy diagram with the profitability of investment on the vertical axis. The horizontal axis measures the proportion of all investors who choose the high-investment strategy, which varies from a minimum of n=zero to a maximum of n=N. The number of investors who choose a low-investment strategy is N-n. An increase in the number of investors choosing a high-investment strategy raises overall investment, and that stimulates demand, which in turn increases the profitability of investment. For the high-investment strategy, the payoff increases more rapidly. Thus, in Figure 1, the thick gray line ab gives the profitability of the high-investment strategy, and the cross-hatched line fg gives the profitability of the low-investment strategy.
An economist naturally assumes that something important happens when two lines cross. In this case y, the proportion at which the two lines cross, is a watershed rather than an equilibrium. Suppose that, at a given time, the number of investors choosing the high-investment strategy is greater than (to the right of) y, but less than N. This is not an equilibrium, since the profit-maximizing strategy for all investors is then the high-investment strategy. On the other hand, suppose that the number of investors choosing the high-investment strategy is less than (to the left of) y, but positive. This is not an equilibrium either, since the profit-maximizing strategy for all investors is then the low-investment strategy. There are 3 equilibria. If all investors choose the high-investment strategy, then we are at the right-hand extreme of the diagram, and the high-investment strategy is the profitable one -- so this is an equilibrium. Similarly, if all investors choose the low-investment strategy, we are at the left extreme of the diagram, and the low-investment strategy is the profitable choice, and this is an equilibrium. Finally, when exactly y investors choose the high-investment strategy, both strategies are equally profitable, so there is no reason for anyone to deviate from it. This, too, is an equilibrium, but it is an unstable one -- any random deviation from it will lead on to one of the extremes.[2]
This example illustrates some key points. Once again we have two equilibria -- and one is better than the other. In the high-investment equilibrium, profits are higher for all investors. The logic behind the example is that the investment community sometimes settles into the low-investment equilibrium, though. When that happens we have a depression or long-term stagnation.
However, there is still a good deal going on behind the scenes. We have said that higher overall investment leads to higher profits, because it stimulates demand. That is a Keynesian idea. But the Keynesian tradition tells us that there are several other sources of spending that stimulate demand. To take them into account we would need to look at things from a more traditionally Keynesian perspective. We will reserve that to an appendix, though. The appendix links these ideas to those in the macroeconomic principles textbook, so it may be of interest to students who have studied macroeconomics; but it is not, in itself, game theory.
Roger A. McCain 