The paper by Robert J. Aumann explores the concepts of subjectivity and correlation in randomized strategies within game theory. Subjectivity refers to the use of subjective probabilities in mixed strategies, while correlation involves the use of correlated strategies. Aumann demonstrates that these concepts can significantly alter the outcomes of games, particularly in non-cooperative settings. Key findings include: 1. **Subjective Strategies in Zero-Sum Games**: In two-person zero-sum games, the introduction of subjective probabilities can lead to cooperative behavior and higher payoffs for both players compared to strictly competitive outcomes. 2. **New Equilibrium Points**: In certain n-person games, new equilibrium points can emerge that dominate all other equilibrium points, even if there is only a small area of subjective disagreement among players. 3. **Correlated Strategies**: Correlated strategies can achieve payoffs that are outside the convex hull of Nash equilibrium payoffs, improving the payoffs for all players. 4. **Formal Model**: A formal model is developed to incorporate subjectivity and correlation, allowing for the analysis of various strategic scenarios. 5. **Equilibrium Points in Two-Person Games**: In two-person games, the introduction of subjective strategies does not necessarily lead to higher payoffs compared to objective mixed strategies. 6. **Two-Person Zero-Sum Games**: In two-person zero-sum games, the introduction of subjective probabilities can still lead to higher payoffs for both players, provided certain conditions are met. Aumann's work highlights the importance of considering both subjectivity and correlation in game theory, providing new insights into the dynamics of strategic interactions.The paper by Robert J. Aumann explores the concepts of subjectivity and correlation in randomized strategies within game theory. Subjectivity refers to the use of subjective probabilities in mixed strategies, while correlation involves the use of correlated strategies. Aumann demonstrates that these concepts can significantly alter the outcomes of games, particularly in non-cooperative settings. Key findings include: 1. **Subjective Strategies in Zero-Sum Games**: In two-person zero-sum games, the introduction of subjective probabilities can lead to cooperative behavior and higher payoffs for both players compared to strictly competitive outcomes. 2. **New Equilibrium Points**: In certain n-person games, new equilibrium points can emerge that dominate all other equilibrium points, even if there is only a small area of subjective disagreement among players. 3. **Correlated Strategies**: Correlated strategies can achieve payoffs that are outside the convex hull of Nash equilibrium payoffs, improving the payoffs for all players. 4. **Formal Model**: A formal model is developed to incorporate subjectivity and correlation, allowing for the analysis of various strategic scenarios. 5. **Equilibrium Points in Two-Person Games**: In two-person games, the introduction of subjective strategies does not necessarily lead to higher payoffs compared to objective mixed strategies. 6. **Two-Person Zero-Sum Games**: In two-person zero-sum games, the introduction of subjective probabilities can still lead to higher payoffs for both players, provided certain conditions are met. Aumann's work highlights the importance of considering both subjectivity and correlation in game theory, providing new insights into the dynamics of strategic interactions.