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博弈论
12
engines
Prisoner's dilemma · the 2×2 payoff bimatrix
Prisoner's dilemma · the 2×2 payoff bimatrix
2×2 T>R>P>S · 纳什(背叛,背叛) vs 帕累托(合作,合作)
Defecting is individually rational — yet mutual cooperation beats it for BOTH. That gap is the dilemma.
Stag hunt · coordination, payoff- vs risk-dominance
Stag hunt · coordination, payoff- vs risk-dominance
协调博弈 · 两纯纳什 · 猎鹿收益占优/猎兔风险占优
Hunt the Stag together for the best outcome — or take the safe Hare alone. Two equilibria, one dilemma of trust.
Public-goods game · free-riding vs the social optimum
Public-goods game · free-riding vs the social optimum
n 人出资 · 乘系数均分 · 搭便车 · 社会最优 vs 纳什0
Everyone gains most if all contribute — but each player privately gains most by contributing nothing.
Pure-strategy Nash equilibrium · 2×2 game
Pure-strategy Nash equilibrium · 2×2 game
2×2 最优反应 · 互为最优反应格=纯纳什 · 0/1/2 个
Best-response analysis — a cell where BOTH players best-respond is a pure Nash equilibrium
Dominant strategy · iterated elimination
Dominant strategy · iterated elimination
严格/弱占优 · 迭代剔除被占优策略 · 求解
Strict dominance and IESDS — strip away dominated strategies until a solution remains
Mixed-strategy Nash equilibrium · 2×2
Mixed-strategy Nash equilibrium · 2×2
无纯均衡(猜硬币) · 无差异概率 p* · 期望收益 vs p
The indifference principle — find the mix that leaves your opponent unable to do better
Minimax & maximin · zero-sum matrix game
Minimax & maximin · zero-sum matrix game
maximin vs minimax · 行最小/列最大 · 鞍点=博弈值
The row player maximizes the worst case; the column player minimizes the best case; they meet at a saddle point
Zero-sum payoff matrix · security levels & value
Zero-sum payoff matrix · security levels & value
所得=所失 · 博弈值 · 鞍点 · 安全水平
Every cell is a pair (row's gain, column's loss) that sums to zero; a saddle point is the value of the game
Pareto frontier · efficient vs dominated outcomes
Pareto frontier · efficient vs dominated outcomes
收益点集 · 非被支配集=前沿 · 支配关系
An outcome is Pareto-efficient when no other is at least as good for both and strictly better for one
Iterated Prisoner's Dilemma · Tit-for-Tat
Iterated Prisoner's Dilemma · Tit-for-Tat
重复囚徒困境 · 先合作后复制 · TFT vs 全背叛/全合作
Pick two fixed strategies and watch the match play out round by round
Evolutionarily Stable Strategy · Hawk–Dove
Evolutionarily Stable Strategy · Hawk–Dove
鹰鸽博弈 · 复制动态 · 稳定鹰比例 p*=V/C
Slide the Hawk fraction and watch fitness drive the population to the stable p*
The Ultimatum Game · rational vs fair
The Ultimatum Game · rational vs fair
提议分配 · 接受/拒绝(拒则双方0) · 子博弈完美 vs 公平阈值
Offer a split of the pie and compare a self-interested responder with a fairness-driven one