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## Managerial Game Theory

Sat, 19 Nov 2011 12:56:01 GMT

One of the first models covered in game theory is the Prisoner’s Dilemma. I’m sure you’ve heard of it or at least recognize it from episodes of Law and Order. Let’s look at the simple model with 2 individual players so we can eliminate mixed strategy equilibria and focus strictly on pure strategies.

The two strategies for each player are “productive” and “exploitive”. The payoffs for the two players can be summarized on the usual grid.

 Employee Productive Exploitive Manager Productive 1,1 0,2 Exploitive 2,0 0,0

One possible economic interpretation of this payoff structure is the ratio of compensation to work output. A ‘1’ represents compensation that accurately prices the employee’s output. A ‘2’ or a ‘0’ represent either excessive compensation for the same output or excessive output for the same compensation. Consistent with these payoffs, the manager prefers to get as much output as possible for as little compensation as possible while the employee prefers the opposite.

In a single period game, the dominant strategy of both players is to choose the exploitive relationship but if both players believe that they will repeat the game, the productive strategy produces the maximum utility for both players. It is fairly safe to assume that both players will begin with the productive strategy since they believe that the game will be played multiple times. The utility of one player starting off with an exploitive relationship and playing until time ‘t’ would be

U= Σ[2, 0/(1+r), 0/(1+r)2, 0/(1+r)3,…0/(1+r)t]

where r is the discount rate under which the player values the next period payoff in the current period. As we can see, the utility of this player is 2 so this player will only begin with an exploitive relationship if he/she believes that

2 > Σ[1, 1/(1+r), 1/(1+r)2, 1/(1+r)3, 0/(1+r)k…0/(1+r)t]

where k+1 is the number of times the game is played before the other player switches to an exploitive strategy. Without an extremely high discount rate ‘r’ the player needs to trust that the other will not switch strategies for relatively few game iterations. So few iterations are needed that I would say that any player with a rational discount rate would start with a productive relationship. Unfortunately, it does not follow that both players will continue to pursue a productive relationship beyond the first few iterations of the game.

The player with the higher discount rate will switch strategies first in order to realize a higher present value of the increased payoff of ‘2’. At this point, both players will pursue the exploitive relationship until the end of the game. Since the discount rate may be interpreted as a measure of risk aversion, the player with greater risk aversion will be the first to pursue an exploitive relationship.

While there are many real world complications with the enforcement of the payoff structure and possible reputational effects for future working relationships, it is important to understand the ease with which reputational effects may be mitigated through HR disclosure policies. Excessive faith in employee controls and risk management policies can lead to risk averse managers and employees both pursuing exploitive relationships. Employees and managers may want to engage in working relationships with riskier counterparties if they want to remain productive.