Question

Consider the general \(2 \times 2\) matrix game: B: \(\quad B_{1} \quad B_{2}\) A: \(\begin{array}{lll}\mathrm{A}_{1} & \mathrm{a}_{11} & \mathrm{a}_{12}\end{array}\) $$ \begin{array}{lll} \mathrm{A}_{2} & \mathrm{a}_{21} & \mathrm{a}_{22} \end{array} $$ Show that if this game has a saddle point, it must have a dominance.

Short answer

By assuming a saddle point exists in a 2x2 matrix game, we showed that there must be a dominance present in either the row or the column containing the saddle point. First, we identified the saddle point using the minimax value \(a_{ij}\). Then, we proved the existence of dominance by considering two cases: row dominance and column dominance. In both cases, we showed inequalities that confirm the dominance: \(a_{ij} \ge a_{kj}\) for row dominance and \(a_{ij} \le a_{il}\) for column dominance.

Step-by-step solution

Identifying a saddle point

First, let's assume that the game has a saddle point. A saddle point occurs where the maximizer player's minimum payoff in a row is equal to the minimizer player's maximum payoff in a column. Accordingly, let's find a_ij for a saddle point in the matrix as follows: \(a_{ij} = min(a_{11}, a_{12}) = max(a_{21}, a_{22}) \)

Proving the existence of dominance

Since we assume that the game has a saddle point, a_ij is the minimax value of the game. Now we will consider any other element(s) in the row and column containing a_ij. There are two cases - either there is dominance at the row level or dominance at the column level.

Case 1: Row dominance.

Suppose row i dominates row k ≠ i. Then for each j, we have: \( a_{ij} \ge a_{kj} \) Now since a_ij is a saddle point and minimax value, we have: \( a_{ij} = min(a_{i1}, a_{i2}) \ge min(a_{k1}, a_{k2}) \) Thus, for at least one j, we have: \( a_{ij} \ge a_{kj} \) Hence, we have row dominance.

Case 2: Column dominance.

Suppose column j dominates column l ≠ j. Then for each i, we have: \( a_{ij} \le a_{il} \) Now since a_ij is a saddle point and minimax value, we have: \( a_{ij} = max(a_{1j}, a_{2j}) \le max(a_{1l}, a_{2l}) \) Thus, for at least one i, we have: \( a_{ij} \le a_{il} \) Hence, we have column dominance. #Conclusion#: By proving the existence of either row or column dominance in the two cases, we have shown that if a 2x2 matrix game has a saddle point, there must be a dominance in the game.

Key concepts

Matrix Games

Matrix games form the foundation of game theory, focusing on interaction between two players. Each player has a set of strategies that determine the outcomes shown in a matrix. In a typical game, one player selects a row, and the other player chooses a column. The intersection of these choices determines the payoff for both players.

To better understand matrix games:

• They are two-player zero-sum games, where one player's gain is exactly balanced by the other's loss.
• The payoff matrix helps visualize strategies and outcomes.
• Each cell in the matrix represents the outcome if certain strategies are chosen.

In a \(2 \times 2\) matrix game, there are two strategies for each player, resulting in a grid with four possible outcomes.

Saddle Point

The saddle point in a matrix game is a crucial concept, serving as a stable point where both players' strategies intersect optimally. This occurs when a value in the matrix is the lowest in its row and the highest in its column.

Key features of a saddle point include:

• The strategy pair at the saddle point is optimal for both players, ensuring the best minimum payoff.
• A matrix can have at most one saddle point, but sometimes none at all.
• It guarantees that neither player can benefit by changing strategies unilaterally.

Identifying a saddle point requires finding the element where its value is both the greatest in its column and the least in its row.

Dominance

Dominance simplifies matrix games by allowing players to ignore less advantageous strategies. A strategy dominates another if it always results in a better (or at least equal) outcome, irrespective of the opponent's choice.

Dominance can be applied in two forms:

• Row Dominance: One row's elements are greater than or equal to those of another row.
• Column Dominance: One column's elements are less than or equal to those of another column.

By identifying such patterns of dominance, players can streamline their strategies, reducing the game matrix and focusing on more impactful decisions. It's especially crucial when seeking the optimal strategy in the presence of a saddle point.

Minimax Theorem

The minimax theorem is a central concept in matrix games, connecting the idea of equilibrium with game strategy optimization. This theorem states that in a zero-sum game, the maximum loss a player can incur is minimized by choosing an optimal strategy.

Main points about the minimax theorem include:

• It ensures that players look beyond immediate gains, considering potential losses.
• Both players operate under the assumption that the other will play their best strategy.
• The minimax value aligns with a saddle point if it exists, representing the game's optimal outcome.

By anchoring gameplay in the perspectives of minimization and maximization, the minimax theorem provides a robust framework for decision-making, foundational to game theory and matrix games.