| Roger McCain's Game Theory: A Nontechnical Introduction to the Analysis of Strategy
(Revised Edition) is available in Asia now and will be available in
USA, UK, Europe and the rest of the World in October. Should you
require the title for fall term, special arrangements can be made so
please do contact Ms Hooi-Yean Lee (mkt@wspc.com).
If you are considering to use this title after fall and would like to
receive an inspection copy, please send your request via World Scientific Publishing Company's website. You may also visit http://www.worldscibooks.com/economics/7517.html for sample chapters. Meanwhile, at a more advanced level, give some thought to McCain's Game Theory and Public Policy, Elgar, 2010. Part 1, "A Historical and Critical Survey," assesses the literature of game theory particularly from the point of view of the pragmatic purposes of public policy, while Part 2 presents some proposals and research toward a game-theoretic foundation for political economy. |
As an illustration of the concepts of sequential games and subgame perfect equilibrium, we shall consider a case in the employment relationship. This game will be a little richer in possibilities than the economics textbook discussion of the supply and demand for labor, in that we will allow for two dimensions of work the principles course does not consider: variable effort and the emotional satisfactions of "meaningful work." We also allow for a sequence of more or less reliable commitments in the choice of strategies.
We consider a three-stage game. At the first stage, one player in the game, the "worker," must choose between two kinds of strategies, that is, two "jobs." In either job, the worker will later have to choose between two rates of effort, "high" and "low." In either job, the output is 20 in the case of high effort and 10 if effort is low. We suppose that the first job is a "meaningful job," in the sense that it meets needs with which the worker sympathizes. As a consequence of this, the worker "feels the pain" of unmet needs when her or his output falls below the potential output of 20. This reduces her or his utility payoff when she or he shirks at the lower effort level. Of course, her or his utility also depends on the wage and (negatively) on effort. Accordingly, in Job 1 the worker's payoff is
wage - 0.3(20-output) - 2(effort)
where effort is zero or one. The other job is "meaningless," so that the worker's utility does not depend on output, and in this job it is
wage - 2(effort)
At the second stage of the game the other player, the "employer," makes a commitment to pay a wage of either 10 or 15. Finally, the worker chooses an effort level, either 0 or 1.
The payoffs are shown in Table 17-1.
|
Job |
||||||
| 1 | 2 | |||||
| effort | 0 | 1 | 0 | 1 | ||
| wage | high | -5, 12 | 5, 13 | -5,15 | 5,13 | |
| low | 0,7 | 10,8 | 0,10 | 10,8 | ||
In each cell of the matrix, the worker's payoff is to the right of the comma and the employer's to the left. Let us first see what is "efficient" here. The payoffs are shown in Figure 1. Payoffs to the employer are on the vertical axis and those to the worker on the horizontal axis. Possible payoff pairs are indicated by stars-of-David. In economics, a payoff pair is said to be "efficient," or equivalently, "Pareto-optimal," if it is not possible to make one player better off without making the other player worse off. The pairs labeled A, B, and C have that property. They are (10,8), (5,13) and (-5,15). The others are inefficient. The red line linking A, B, and C is called the utility possibility frontier. Any pairs to the left of and below it are inefficient.
Now let us explore the subgame perfect equilibrium of this model. First, we may see that the low wage is a "dominant strategy" for the employer. That is, regardless which strategy the worker chooses -- job 1 and low effort, job 2 and high effort, and so on -- the employer is better off with low wages than with high. Thus the worker can anticipate that the wages will be low. Let us work backward. Suppose that the worker chooses job 2 at the first stage. This limits the game to the right-hand side of the table, which has a structure very much like the Prisoners' Dilemma. In this subgame, both players have dominant strategies. The worker's dominant strategy is low effort, and the Prisoners' Dilemma-like outcome is at (0,10). This is the outcome the worker must anticipate if he chooses Job 2.
What if he chooses Job 1? Then the game is limited to the left-hand side. In this game, too, the worker, like the employer, has a dominant strategy, but in this case it is high effort. This subgame is not Prisoners' Dilemma-like, since the equilibrium -- (10,8) -- is an efficient one. This is the outcome the worker must expect if she or he chooses Job 1, "meaningful work."
But the worker is better off in the subgame defined by "nonmeaningful work," Job 2. Accordingly, she will choose Job 2, and thus the equilibrium of the game as a whole (the subgame perfect equilibrium) is at (0,10). It is indicated by point E in the figure, and is inefficient.
Why is meaningful work not chosen in this model? It is not chosen because there is no effective reward for effort. With meaningful work, the worker can make no higher wage, despite her greater effort. Yet she does not reduce her effort because doing so brings the greater utility loss of seeing the output of meaningful work decline on account of her decision. The dilemma of having to choose between a financially unrewarded extra effort and witnessing human suffering on account of one's failure to make the effort seems to be a very stylized account of what we know as "burnout" in the human service professions.
Put differently, workers do not choose meaningful work at low wages because they have a preferable alternative: shirking at low effort levels in nonmeaningful jobs. Unless the meaningful jobs pay enough to make those jobs, with their high effort levels, preferable to the shirking alternative, no-one will choose them.
Inefficiency in Nash equilibria is a consequence of their noncooperative nature, that is, of the inability of the players to commit themselves to efficiently coordinated strategies. Suppose they could do so -- what then? Suppose, in particular, that the employer could commit herself or himself, at the outset, to pay a high wage, in return for the worker's commitment to choose Job 1. There is no need for an agreement about effort -- of the remaining outcomes, in the upper left corner of the table, the worker will choose high effort and (5,13), because of the "meaningful" nature of the work. This is an efficient outcome.
And that, after all, is the way markets work, isn't it? Workers and employers make reciprocal commitments that balance the advantages to one against the advantages to the other? It is, of course, but there is an ambiguity here about time. There is, of course, no measurement of time in the game example. But commitments to careers are lifetime commitments, and correspondingly, the wage incomes we are talking about must be lifetime incomes. The question then becomes, can employers make credible commitments to pay high lifetime income to workers who choose "meaningful" work with its implicit high effort levels? In the 1960's, it may have seemed so; but in 1995 it seems difficult to believe that the competitive pressures of a profit-oriented economic system will permit employers to make any such credible commitments.
This may be one reason why "meaningful work" has generally been organized through nonprofit agencies. But under present political and economic conditions, even those agencies may be unable to make credible commitments of incomes that can make the worker as well off in a high-effort meaningful job as in a low-effort nonmeaningful one. If this is so, there may be little long-term hope for meaningful work in an economy dominated by the profit system.
Lest I be misunderstood, I do not mean to argue that a state-organized system would do any better. There is an alternative: a system in which worker incomes are among the objectives of enterprises, that is, a cooperative system. It appears to be possible that such a system could generate meaningful work. There is empirical evidence that cooperative enterprises do in fact support higher effort levels than either profit-oriented or state organizations.
Of course, some nonmeaningful work has to be done, and it remains true that when nonmeaningful work is done it is done inefficiently and at a low effort level, that is, at E in the figure. In other words, the fundamental source of inefficiency in this model is the inability of the workers to make a credible commitment to high effort levels. If high effort could somehow be assured, then (depending on bargaining power) a high-effort efficient outcome would become a possibility in the nonmeaningful work subgame, and this in turn would eliminate the worker's incentive to choose nonmeaningful work in order to shirk. (If worker bargaining power should enforce the outcome at C, which is Pareto-optimal, the shirking nonmeaningful strategy would still dominate meaningful work). However, it does seem that it is very difficult to make commitments to high effort levels credible, or enforceable, in the context of profit-oriented enterprises.
It may be, then, the the problem of finding meaningful work and of burn-out in fields of meaningful work is a relatively minor aspect of the far broader question of effort commitment in modern economic systems. Perhaps it will do nevertheless as an example of the application of subgame perfect equilibrium concepts to an issue of considerable interest to many modern university students.
Roger A. McCain 