Define Two Person Zero Sum Game

In game theory, the concept of a two-person zero-sum game provides a foundational framework for understanding competitive interactions. This model, while simplified, offers valuable insights into scenarios where the gains of one participant directly correlate with the losses of the other, creating a strictly adversarial environment.

Understanding Zero-Sum Games: The Basics

At its core, a zero-sum game is a situation where the total gains of all players, minus the total losses, equals zero. This means that wealth or benefit is simply transferred from one player to another; no value is created or destroyed. Imagine a pie of a fixed size: when one person takes a larger slice, the remaining slices must necessarily be smaller.

Key Characteristics:

• Two Players: The model focuses on interactions between only two participants.
• Conflicting Interests: Players have diametrically opposed goals; what benefits one harms the other.
• Fixed Resources: The total available resources or benefits are constant.
• Zero-Sum Condition: The sum of payoffs for all players always equals zero. If one player wins an amount 'x', the other player loses the same amount 'x'.

Formal Definition of a Two-Person Zero-Sum Game

A two-person zero-sum game can be formally defined as a game where:

1. There are two players, traditionally referred to as Player A and Player B.
2. Each player has a set of possible strategies. Let's say Player A has m strategies and Player B has n strategies.
3. A payoff matrix defines the outcome for each combination of strategies. The payoff matrix represents the amount Player A wins from Player B (or, equivalently, the amount Player B loses to Player A).
4. The sum of the payoffs for Player A and Player B for any given strategy profile (combination of strategies) is always zero. Mathematically, this can be represented as: Payoff to Player A + Payoff to Player B = 0. Or, Payoff to Player A = - Payoff to Player B.

This definition highlights the critical element: the strict interdependence of the players' outcomes. One player's success is inherently tied to the other player's failure.

Illustrative Examples of Two-Person Zero-Sum Games

To solidify the understanding, let's explore some practical examples:

• Chess: Chess is a classic example. One player wins, the other loses, or it's a draw. If we assign +1 to a win for Player A, -1 to a win for Player B, and 0 for a draw (which can be split as 0 for each player), the sum is always zero.
• Heads or Tails: Two players flip a coin. One player calls "heads," the other calls "tails." If the coin lands on heads, the "heads" player wins a dollar from the "tails" player. If it lands on tails, the "tails" player wins a dollar. The gain of one is precisely the loss of the other.
• Competitive Bidding: In a sealed-bid auction where only one item is being sold, and the item has a fixed value (or an agreed-upon value), the situation can be modeled as a zero-sum game. The winner gains the item (which we can normalize to a value of zero since everyone agrees on its value) but pays their bid. The loser loses their bid and gains nothing. The net result is the transfer of money from the loser to the winner.
• Penalty Shootout in Soccer: In a penalty shootout, each team alternates taking penalty kicks. The team that scores more goals wins. We can model this as a zero-sum game where the payoffs are based on the probability of winning given a specific sequence of made and missed kicks.

Examples That Aren't Zero-Sum Games

It is equally crucial to understand scenarios that do not fit the zero-sum model:

• Most Economic Transactions: A typical business transaction is not zero-sum. When you buy a coffee, you value the coffee more than the money you paid, and the coffee shop values the money more than the coffee. Both parties benefit. This is a positive-sum game.
• Negotiations Leading to Agreement: If two companies negotiate a mutually beneficial partnership, it's not zero-sum. Both companies stand to gain from the collaboration.
• Cooperative Games: Games where players can form coalitions and cooperate to achieve a common goal are generally not zero-sum. The total payoff can be increased through cooperation.

Mathematical Representation: The Payoff Matrix

The payoff matrix is a crucial tool for analyzing two-person zero-sum games. It provides a clear representation of the outcomes for each player based on the strategies they choose.

Consider a simple game where Player A has two strategies (A1 and A2) and Player B also has two strategies (B1 and B2). The payoff matrix would look like this:

In this matrix:

• x represents the payoff to Player A if Player A chooses strategy A1 and Player B chooses strategy B1. Consequently, -x is the payoff to Player B.
• y represents the payoff to Player A if Player A chooses strategy A1 and Player B chooses strategy B2. Consequently, -y is the payoff to Player B.
• z represents the payoff to Player A if Player A chooses strategy A2 and Player B chooses strategy B1. Consequently, -z is the payoff to Player B.
• w represents the payoff to Player A if Player A chooses strategy A2 and Player B chooses strategy B2. Consequently, -w is the payoff to Player B.

Example:

Let's say the payoff matrix is:

This means:

• If Player A chooses A1 and Player B chooses B1, Player A wins 2 units from Player B.
• If Player A chooses A1 and Player B chooses B2, Player A loses 1 unit to Player B.
• If Player A chooses A2 and Player B chooses B1, Player A loses 3 units to Player B.
• If Player A chooses A2 and Player B chooses B2, Player A wins 4 units from Player B.

Solving Two-Person Zero-Sum Games: Finding Optimal Strategies

The goal of each player in a two-person zero-sum game is to maximize their own payoff, knowing that the other player is trying to minimize it. This leads to the concept of optimal strategies. There are several approaches to finding these strategies:

1. Dominance

Dominance occurs when one strategy consistently yields a better outcome for a player, regardless of the opponent's strategy. A dominant strategy is always preferable and should be chosen.

• Strict Dominance: Strategy A strictly dominates strategy B if, for every strategy of the opponent, the payoff from A is strictly greater than the payoff from B.
• Weak Dominance: Strategy A weakly dominates strategy B if, for every strategy of the opponent, the payoff from A is greater than or equal to the payoff from B, and for at least one strategy of the opponent, the payoff from A is strictly greater than the payoff from B.

Example:

Consider the following payoff matrix for Player A:

In this case, strategy A1 strictly dominates strategy A2 because, regardless of whether Player B chooses B1 or B2, Player A gets a higher payoff by choosing A1. Therefore, Player A should always choose A1. Player B, realizing this, will choose the strategy that minimizes Player A's gain, which is B2 in this case.

2. Saddle Point (Pure Strategy Equilibrium)

A saddle point is a cell in the payoff matrix that represents the minimum value in its row and the maximum value in its column. If a saddle point exists, it represents a pure strategy equilibrium. This means both players have a single, best strategy that they should always play.

Finding the Saddle Point:

1. For each row, find the minimum value.
2. For each column, find the maximum value.
3. If there is a cell that is both the minimum of its row and the maximum of its column, that is the saddle point.

Example:

• Row minimums: Row 1: 2, Row 2: 4
• Column maximums: Column 1: 6, Column 2: 4

The value 4 is the minimum of its row (Row 2) and the maximum of its column (Column 2). Therefore, there is a saddle point at (A2, B2). Player A should always play A2, and Player B should always play B2. The value of the game is 4 (for Player A).

Important Note: Not all two-person zero-sum games have a saddle point.

3. Mixed Strategies

When a game does not have a saddle point, players need to employ mixed strategies. A mixed strategy involves randomly choosing between different strategies with specific probabilities. This introduces an element of unpredictability, making it harder for the opponent to exploit a fixed pattern.

Finding Optimal Mixed Strategies:

There are several methods for finding optimal mixed strategies, including:

• Graphical Method (for 2x2 games): This method is suitable for games where both players have only two strategies. It involves plotting the expected payoff for one player as a function of the probability of the other player using one of their strategies. The optimal mixed strategy is found at the intersection of the lines representing the expected payoffs.
• Algebraic Method (for 2x2 games): This method involves setting up equations to find the probabilities that make the opponent indifferent between their strategies. The goal is to find probabilities that equalize the expected payoff for the opponent, regardless of which strategy they choose.
• Linear Programming: This is a more general method that can be used for games with any number of strategies. It involves formulating the problem as a linear program and using optimization techniques to find the optimal mixed strategies.

Example (Algebraic Method for a 2x2 game):

Consider the following payoff matrix for Player A:

Let p be the probability that Player A chooses strategy A1, and (1-p) be the probability that Player A chooses strategy A2.

Let q be the probability that Player B chooses strategy B1, and (1-q) be the probability that Player B chooses strategy B2.

For Player B to be indifferent between their strategies, the expected payoff for Player A must be the same regardless of whether Player B chooses B1 or B2:

4p + 2*(1-p) = 1p + 3*(1-p)

Solving for p:

4p + 2 - 2p = p + 3 - 3p 2p + 2 = -2p + 3 4p = 1 p = 1/4

Therefore, Player A should choose A1 with probability 1/4 and A2 with probability 3/4.

Similarly, for Player A to be indifferent between their strategies, the expected loss for Player B must be the same regardless of whether Player A chooses A1 or A2:

• (4q + 1*(1-q)) = - (2q + 3*(1-q)) 4q + 1 - q = 2q + 3 - 3q 3q + 1 = -q + 3 4q = 2 q = 1/2

Therefore, Player B should choose B1 with probability 1/2 and B2 with probability 1/2.

The value of the game for Player A can be calculated by substituting p into either of the expected payoff equations for Player A (or q into the expected loss equations for Player B):

4*(1/4) + 2*(3/4) = 1 + 3/2 = 5/2 = 2.5

The value of the game is 2.5, meaning that on average, Player A will win 2.5 units from Player B if both players play their optimal mixed strategies.

Real-World Applications and Limitations

While the two-person zero-sum game model is a simplification of reality, it provides a useful framework for analyzing competitive situations.

Applications:

• Negotiations: Certain aspects of negotiations, particularly those involving dividing fixed resources, can be modeled as zero-sum games. Understanding the dynamics can help negotiators develop effective strategies.
• Competitive Sports: Many competitive sports, especially head-to-head matchups, exhibit zero-sum characteristics. Analyzing strategies in these contexts can improve performance.
• Military Strategy: In some military scenarios, the objective is to inflict maximum damage on the enemy while minimizing one's own losses, which can be framed as a zero-sum game.
• Resource Allocation: Situations involving the allocation of scarce resources among competing entities can sometimes be modeled as zero-sum games.

Limitations:

• Oversimplification: The assumption of perfect antagonism and fixed resources rarely holds true in real-world scenarios. Most situations are more complex and involve elements of cooperation or the possibility of creating new value.
• Ignoring External Factors: The model focuses solely on the interaction between two players, ignoring the influence of external factors or third parties.
• Rationality Assumption: The model assumes that players are perfectly rational and will always act in their own best interests. This is not always the case in reality, as emotions, biases, and incomplete information can influence decision-making.
• Difficulty in Quantifying Payoffs: Accurately quantifying the payoffs for each player can be challenging, especially in complex situations where the outcomes are uncertain or difficult to measure.

Variations and Extensions

The basic two-person zero-sum game model has been extended in several ways to address some of its limitations:

• N-Person Zero-Sum Games: These models extend the zero-sum concept to games with more than two players. While the total gains still equal the total losses, the analysis becomes more complex due to the possibility of coalitions and strategic alliances.
• General-Sum Games: These models allow for the possibility that the sum of payoffs for all players is not zero. This captures situations where value can be created or destroyed, leading to outcomes that are mutually beneficial (positive-sum) or mutually destructive (negative-sum).
• Repeated Games: These models analyze the effects of repeated interactions between players. In repeated games, players can learn from past experiences and adjust their strategies accordingly, leading to more complex and nuanced outcomes.

Conclusion

The two-person zero-sum game is a fundamental concept in game theory that provides a valuable framework for understanding competitive interactions where one player's gain is directly equivalent to the other player's loss. While it simplifies reality by assuming perfect antagonism and fixed resources, it offers important insights into strategic decision-making in various fields, from negotiations and sports to military strategy and resource allocation. By understanding the key characteristics, mathematical representation, and solution methods for two-person zero-sum games, individuals and organizations can better navigate competitive environments and develop strategies to maximize their chances of success. Though the model has limitations, its core principles remain relevant as a building block for understanding more complex game-theoretic scenarios. It is vital to remember that real-world situations are rarely purely zero-sum, and the principles learned from this model should be applied with careful consideration of the broader context and potential for cooperation or value creation.