Zero-Sum Fictitious Play Cannot Converge to a Point
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911577479839744 |
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| author | Moon, Jaehong |
| author_facet | Moon, Jaehong |
| contents | Fictitious play (FP) is a history-based strategy to choose actions in normal-form games, where players best-respond to the empirical frequency of their opponents' past actions. While it is well-established that FP converges to the set of Nash equilibria (NE) in zero-sum games, the question of whether it converges to a single equilibrium point, especially when multiple equilibria exist, has remained an open challenge.
In this paper, we establish that FP does not necessarily stabilize at a single equilibrium. Specifically, we identify a class of zero-sum games where pointwise convergence fails, regardless of the tie-breaking rules employed. We prove that two geometric conditions on the NE set (A1 and A2) and a technical assumption (A3) are sufficient to preclude pointwise convergence. Furthermore, we conjecture that the first two conditions alone may be sufficient to guarantee this non-convergence, suggesting a broader fundamental instability in FP dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_07544 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Zero-Sum Fictitious Play Cannot Converge to a Point Moon, Jaehong Computer Science and Game Theory Fictitious play (FP) is a history-based strategy to choose actions in normal-form games, where players best-respond to the empirical frequency of their opponents' past actions. While it is well-established that FP converges to the set of Nash equilibria (NE) in zero-sum games, the question of whether it converges to a single equilibrium point, especially when multiple equilibria exist, has remained an open challenge. In this paper, we establish that FP does not necessarily stabilize at a single equilibrium. Specifically, we identify a class of zero-sum games where pointwise convergence fails, regardless of the tie-breaking rules employed. We prove that two geometric conditions on the NE set (A1 and A2) and a technical assumption (A3) are sufficient to preclude pointwise convergence. Furthermore, we conjecture that the first two conditions alone may be sufficient to guarantee this non-convergence, suggesting a broader fundamental instability in FP dynamics. |
| title | Zero-Sum Fictitious Play Cannot Converge to a Point |
| topic | Computer Science and Game Theory |
| url | https://arxiv.org/abs/2604.07544 |