In a paper, recently published in the International Journal of Game Theory, my wife and I analyze a game called a forced contribution threshold public good game. A nice way to illustrate the game is to look at the difficulties of getting rid of an incompetent manager.
So, consider a department with n workers who all want to get rid of the manager. If they don't get rid of him then there payoff will be L. If they do get rid of him then there payoff will be H > L. But, how to get rid of him? He will only be removed if at least t or more of the workers complain to senior management. For instance, if a majority of staff need to complain then t = n/2.
If t or more complain then the manager is removed and everyone is happy. The crucial thing, though, is what happens if less than t complain. In this case the manager will remain and any workers that did complain will face recrimination. To be specific suppose that the cost of recrimination is C. Then potential payoffs to a worker called Jack are as follows:
If t or more complain then Jack gets payoff H.
If Jack complains but not enough others do then he gets payoff L - C.
If Jack does not complain and others don't either then he gets payoff L.
Note that this game is called a 'forced' contribution game because, if the manager is removed, Jack's payoff does not depend on whether or not he complained. This contrasts with a standard threshold public good game in which those who do not contribute (i.e. complain) always have a relative advantage. Hence, there is a sense in which every worker is 'forced to contribute' if the manager is removed.
The fear of recrimination is key to the game and going to be the potential source of inefficiency. In particular, if Jack fears that others will not complain then it is not in his interest to complain either. Hence we can obtain an inefficient equilibrium in which nobody complains and the manager carries on before. This is not good for the workers and presumably not good for the firm either. So, how can this outcome be avoided?
In our paper we compare the predictions of three theoretical models and then report an experiment designed to test the respective predictions. Our results suggest that the workers will struggle to get rid of the manager if
This means that the threshold t should not be set too high. For instance, if a simple majority is needed to get rid of the manager, and so t = n/2, then we need H to about 25% higher than L. If less than a majority is enough then H does not need to be as high. This result would suggest that it is relatively simple to have a corporate policy that would incentivise people like Jack to complain about his manager.
Of course, in practice there are almost certainly going to be some who will defend the manager and so things become more complex. Moreover, there are likely to be significant inertia effects. In particular, the 'better the devil you know' attitude may lead workers to underestimate the difference between H and L. Also senior managers may set t relatively high because of a desire to back managers. These are all things that will make it less likely Jack complains and more likely the incompetent manager continues. Firms, therefore, need to strike the right balance to weed out inefficiency.
So, consider a department with n workers who all want to get rid of the manager. If they don't get rid of him then there payoff will be L. If they do get rid of him then there payoff will be H > L. But, how to get rid of him? He will only be removed if at least t or more of the workers complain to senior management. For instance, if a majority of staff need to complain then t = n/2.
If t or more complain then the manager is removed and everyone is happy. The crucial thing, though, is what happens if less than t complain. In this case the manager will remain and any workers that did complain will face recrimination. To be specific suppose that the cost of recrimination is C. Then potential payoffs to a worker called Jack are as follows:
If t or more complain then Jack gets payoff H.
If Jack complains but not enough others do then he gets payoff L - C.
If Jack does not complain and others don't either then he gets payoff L.
Note that this game is called a 'forced' contribution game because, if the manager is removed, Jack's payoff does not depend on whether or not he complained. This contrasts with a standard threshold public good game in which those who do not contribute (i.e. complain) always have a relative advantage. Hence, there is a sense in which every worker is 'forced to contribute' if the manager is removed.
The fear of recrimination is key to the game and going to be the potential source of inefficiency. In particular, if Jack fears that others will not complain then it is not in his interest to complain either. Hence we can obtain an inefficient equilibrium in which nobody complains and the manager carries on before. This is not good for the workers and presumably not good for the firm either. So, how can this outcome be avoided?
In our paper we compare the predictions of three theoretical models and then report an experiment designed to test the respective predictions. Our results suggest that the workers will struggle to get rid of the manager if
Of course, in practice there are almost certainly going to be some who will defend the manager and so things become more complex. Moreover, there are likely to be significant inertia effects. In particular, the 'better the devil you know' attitude may lead workers to underestimate the difference between H and L. Also senior managers may set t relatively high because of a desire to back managers. These are all things that will make it less likely Jack complains and more likely the incompetent manager continues. Firms, therefore, need to strike the right balance to weed out inefficiency.
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