Abstract
We introduce a new fundamental problem called triangular trade, which is a natural extension of the well-studied prisoner’s dilemma for three (or more) players where a player cannot directly punish a seemingly defecting player. More specifically, this problem deals with a situation where the power/influence of players is one-way, players would be better off if they maintain circular cooperation, but each player has an incentive to defect. We analyze whether players can sustain such circular cooperation when they repeatedly play this game and each player observes the actions of another player with some observation errors (imperfect private monitoring). We confirm that no simple strategy can constitute an equilibrium within any reasonable parameter settings when there are only two actions: “Cooperate” and “Defect.” Thus, we introduce two additional actions: “Whistle” and “Punish,” which can be considered as a slight modification of “Cooperate.” Then, players can achieve sustainable cooperation using a simple strategy called Remote Punishment strategy (RP), which constitutes an equilibrium for a wide range of parameters. Furthermore, we show the payoff obtained by a variant of RP is optimal within a very general class of strategies that covers virtually all meaningful strategies.
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Notes
- 1.
We say a strategy is simple when it is concisely represented by a finite-state automaton with a few states.
- 2.
The same applies to action \(a_{i\pm k}\) or state \(\theta _{i\pm k}\).
- 3.
There exist many other directions to extend the PD for three or more players, including the well-known public goods game [13]. Our extension is original, as it addresses the case where a player cannot directly punish a seemingly deviating player.
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Acknowledgements
This work was partially supported by JSPS KAKENHI (Grant Number 16KK0003, 17H00761, and 17H01787) and JST, Strategic International Collaborative Research Program, SICORP.
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Shigedomi, K., Sekiguchi, T., Iwasaki, A., Yokoo, M. (2018). Repeated Triangular Trade: Sustaining Circular Cooperation with Observation Errors. In: Miller, T., Oren, N., Sakurai, Y., Noda, I., Savarimuthu, B.T.R., Cao Son, T. (eds) PRIMA 2018: Principles and Practice of Multi-Agent Systems. PRIMA 2018. Lecture Notes in Computer Science(), vol 11224. Springer, Cham. https://doi.org/10.1007/978-3-030-03098-8_15
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