# Games with incomplete information and uncertain payoff: from the perspective of uncertainty theory

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## Abstract

In a game with incomplete information, players do not possess full information about their opponents. For such a game, Bayesian approaches which are based on probability theory were employed to find the equilibria. Sometimes, players are lack of data for probabilistic reasoning in which case Bayesian methods cannot be used. In this paper, we adopt a new mathematical framework—uncertainty theory to solve such a game. The player’s type (incomplete information) is modeled as an uncertain variable. In addition to previous studies, the payoff for each player is uncertain in our research, which is a realistic assumption in practice. We first define the games with incomplete information with uncertain payoff (Iu-game). Then, we create a new game (U-game) as a method to solve Iu-game. A theorem is provided to show that the equilibria are the same for both games. We use an example to illustrate the application of the proposed theorem. Finally, we generalize the form of U-game and show the equilibria are independent of the incomplete information.

## Keywords

Uncertainty theory Game theory Incomplete information Uncertain equilibrium## Notes

### Acknowledgements

The paper is supported by National Natural Science Foundation of China Nos. 71704007, 61703014, the Base Project of Beijing Social Science Foundation No. 17JDGLB019

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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