Abstract
We consider the repeated minimum-effort coordination game where each player follows an adaptive strategy in each period and his choice is made via the logit probability distribution. We find that there exists a stable probability distribution of the minimum effort levels (called the equilibrium of the game), and the expected value of the minimum effort levels at the equilibrium has the same comparative-statics properties as in the experimental outcomes of Van Huyck et al. (Am Econ Rev 80(1):234–248 1990): it decreases with the effort cost and the number of players. We also find that the expected value at the equilibrium responds differently to the noise parameter, contingent on the effort-cost structure. This provides us with an implication about how we could increase the coordination among the players.
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Notes
Due to the perfect complementarity between players’ efforts, the greater common gain can be achieved only when all the players choose greater effort levels. However, from an individual player’s perspective, the greater gain requires the greater effort level and it entails greater risk, because it is uncertain whether all the other players will choose the greater effort levels or not. As the number of players increases, the uncertainty about the other players’ choices increases, which implies that the expected gain decreases, and thus the players are less likely to choose the greater effort level. Similarly, if the number of players stays constant but the effort cost increases, the uncertainty remains the same, i.e., the expected gain doesn’t change, but the cost for choosing the greater effort level goes up, which means that the expected net gain decreases. Hence, the players are less likely to choose the greater effort level as the effort cost increases.
The parameter values are arbitrary, as an example. In the next section, we will continue our discussion while using the same parameter values taken in the experiment of Goeree and Holt (2005).
We have implemented the Monte Carlo simulations with many other parameter values and confirmed that all simulation results agree with the equilibrium values derived from the proposition.
The payoff for player i in Eq. 2 can be expressed: \(\pi _{i} (e_{i} , m_{t} )=a \left (\min (e_{i} , m_{t}) - \frac {b}{a} e_{i}\right )\), where \(\frac {b}{a}\) shows the ratio between marginal cost for putting effort and marginal (common) benefit, i.e., the cost-benefit ratio. So, if we set a = 1, parameter b comes to the cost-benefit ratio.
Also note that our equilibrium value is derived in a dynamic setting, while the experimental setting is static. So, once we can use the average values of the minimum effort levels instead of the pooled average effort levels observed in the experiment, the difference between our equilibrium values and the experimental results (the average minimum-effort levels) may be caused by the (negative) learning effect. Furthermore, we may get more information about players’ behavior under each cost structures, e.g., whether players decide less rationally in the high-cost structure or not, by using the different fitting values of μ. However, unfortunately, we could not find the relevant data or information in the paper. We ask for the readers’ understanding and thank the anonymous reviewer for the insightful comments in this regard.
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Funding
This study was funded by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2015S1A5A2A03049830).
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This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2015S1A5A2A03049830)
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Kim, P., Lee, D. Repeated minimum-effort coordination games. J Evol Econ 29, 1343–1359 (2019). https://doi.org/10.1007/s00191-018-0587-z
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DOI: https://doi.org/10.1007/s00191-018-0587-z