Problem Set 5 Zero Sum Games Prepared By Joseph Malkevitch D
Problem Set 5 Zero Sum Gamesprepared Byjoseph Malkevitchdepartment
Analyze three zero-sum matrix games (U, V, and W) where payoffs are from the row player's perspective, focusing on the existence of pure strategies, saddle points, dominant strategies, and mixed strategies. Additionally, evaluate the fairness of each game and compare the outcomes of optimal play to equal probability strategies. Finally, extend the analysis to two additional games (X and Y), determining their game values through various methods and examining the implications of their payoff structures.
Paper For Above instruction
Zero-sum games are a fundamental concept in game theory, representing competitive scenarios where one player's gain is exactly balanced by the other's loss. Analyzing such games involves identifying optimal strategies—whether pure or mixed—and determining the value of the game, which signifies the expected payoff under optimal play. This paper provides a comprehensive analysis of three matrix games labeled U, V, and W, exploring their strategic properties, and extends the discussion to two additional games, X and Y.
Analysis of Games U, V, and W
To analyze each game, we examine the payoff matrices to identify the presence of saddle points, determine if pure strategies are sufficient for optimal play, consider dominant strategies, and if necessary, calculate mixed strategy equilibria. The fairness of each game is assessed by comparing the game value to zero, as a game with a zero expected payoff for optimal strategies is considered fair.
Game U: Characteristics and Strategic Analysis
In game U, the payoff matrix is examined for any saddle point—an element that is the smallest in its row and the largest in its column—indicating a pure strategy equilibrium. If such a saddle point exists, the game has a defined value, and the players' optimal strategies are pure. If not, we analyze whether dominant strategies arise for either player. Absent these, we proceed to find mixed strategy solutions using linear programming techniques.
Suppose the matrix for U is:
| Column I | Column II | |
|---|---|---|
| Row 1 | a | b |
| Row 2 | c | d |
By inspecting this matrix, we look for a saddle point—say, if a is minimal in row 1 but maximal in column I, indicating a saddle point at (Row 1, Column I). If such a cell exists, that is the game value, and strategies are pure. If not, we evaluate potential dominance and proceed with mixed strategies to estimate the game value and optimal play.
Game V and W: Similar Analysis
Applying the same methodology, the analysis involves checking for the existence of saddle points, dominances, and calculating mixed strategies via linear programming if necessary. Each game's matrix structure and payoff entries influence whether pure or mixed strategies are optimal and whether the game is fair. A game is considered fair if the value of the game is zero, implying neither player has an advantage under optimal play.
Extensions to Games X and Y: Comparing Outcomes and Strategies
Given the assumptions that these games lack pure strategy solutions, players resort to mixed strategies, often involving flipping coins or probabilistic selection of strategies. The comparison involves examining the payoffs under these mixed strategies versus outcomes if players could identify pure strategies. Typically, the expected payoff under mixed strategies in such games approximates the value of the game obtained through equilibrium analysis.
For game X and Y, the analysis begins with setting up the payoff matrices, then determining whether saddle points exist. If not, strategies are derived through solving linear programming problems aimed at maximizing a player's minimum payoff (for Row) or minimizing the maximum payoff (for Column). The benefits of mixed strategies are evident when no pure strategies yield equilibrium, and decision-making under uncertainty becomes optimal.
Concluding Remarks and Patterns
Throughout these analyses, a recurring pattern emerges: pure strategies suffice when saddle points exist, simplifying decision-making. In their absence, mixed strategies predominate, highlighting the importance of probabilistic reasoning in competitive scenarios. The fairness of games hinges on their value; games with values close to zero are balanced, while positive or negative values indicate advantage or disadvantage for the row player, respectively. Recognizing these patterns assists in understanding strategic complexity within zero-sum contexts and informs real-world applications, such as military strategy, market competition, and political decision-making.
References
- Cournot, A. (1838). Recherches sur les principes mathematiques de la theorie des richesses. Paris: Hachette.