Question

Smith and Jones are playing a number-matching game. Each chooses cither \(1,2,\) or \(3 .\) If the numbers match, Jones pays Smith \(\$ 3 .\) If they differ, Smith pays Jones \(\$ 1\) a. Describe the payoff matrix for this game and show that it does not possess a Nash equilibrium strategy pair. b. Show that with mixed strategies this game does have a Nash equilibrium if each player plays each number with probability \(1 / 3 .\) What is the value of this game?

Short answer

Based on the step-by-step solution provided above, write a short answer as follows: In the number-matching game between Smith and Jones, there is no Nash equilibrium strategy pair in pure strategies. However, there is a Nash equilibrium strategy pair in mixed strategies, where both players play each number with equal probability (Smith: p = q = r = 1/3, Jones: x = y = z = 1/3). The value of this game, to two decimal places, is approximately 0.33 (for Smith).

Step-by-step solution

Describe the Payoff Matrix

To describe the payoff matrix for this game, we need to list the payoffs for each possible combination of choices by Smith and Jones. Let the rows represent Smith's choices and the columns represent Jones' choices. The payoff matrix will look like this: |\(1\)|\(2\)|\(3\)| |---|---|---| |\(1\)|(3,-3)|(-1,1)|(-1,1)| |\(2\)|(-1,1)|(3,-3)|(-1,1)| |\(3\)|(-1,1)|(-1,1)|(3,-3)| Here, each cell contains a pair of payoffs - (Smith's payoff, Jones' payoff).

Check for Nash Equilibrium Strategy Pair in Pure Strategies

A Nash equilibrium strategy pair in pure strategies is a pair of strategies where neither player would want to deviate from their choice given the other player's choice. To find a Nash equilibrium, we will search for any dominant strategies or check each cell to see if neither player would be willing to switch. Looking at the payoff matrix, there are no dominant strategies, and there is no cell where both players would not want to deviate from their choices. So, this game does not have a Nash equilibrium strategy pair in pure strategies.

Determine the Nash Equilibrium Strategy Pair in Mixed Strategies

We will now check if there is a Nash equilibrium in mixed strategies by finding the probabilities with which each player should play each number. Let \(p, q,\) and \(r\) be the probabilities for Smith and \(x, y,\) and \(z\) be the probabilities for Jones, where \(p+q+r=1\) and \(x+y+z=1\). Then the expected payoffs for Smith are as follows: 1 vs 1: \(3x = 3 - 3z\) 1 vs 2: \(-y = -y\) 1 vs 3: \(-z = -z\) 2 vs 1: \(-x=-x\) 2 vs 2: \(3y = 3 - 3x\) 2 vs 3: \(-z = -z\) 3 vs 1: \(-x=-x\) 3 vs 2: \(-y = -y\) 3 vs 3: \(3z = 3 - 3x - 3y\) To find the Nash equilibrium strategy pair, we need to find the probabilities that make the expected payoffs equal for all three possible choices of Smith when facing Jones' mixed strategy. So, we need to find \(p, q,\) and \(r\) that satisfy the following conditions: \(3 - 3z = -y = -z\) \(-x = 3 - 3x = -z\) \(-x = -y = 3 - 3x - 3y\) Solving this system of equations, we find that \(p = q = r = \frac{1}{3}\) and \(x = y = z = \frac{1}{3}\).

Compute the Value of the Game

Now that we have found the Nash equilibrium strategy pair in mixed strategies, we can evaluate the value of the game. To do this, we compute the expected payoff for the players playing according to the Nash equilibrium strategy pair: Expected payoff for Smith = \(\frac{1}{3}\times(3) +\frac{1}{3}\times(-1)+\frac{1}{3}\times(-1)=1-0.3333-0.3333 =0.3333\) The value of this game, to two decimal places, is approximately \(0.33\) (for Smith).

Key concepts

Payoff Matrix

In game theory, the payoff matrix is a table that summarizes the outcomes of a game for each combination of strategies played by the participants. The matrix displays the gains (positive values) or losses (negative values) for each player, based on the choices both players make.

To illustrate, let's consider the payoff matrix of Smith and Jones' game. Each has three choices (1, 2, or 3), and their payoffs differ depending on whether the numbers match or not. When they match, Jones pays Smith \(3, represented by positive values for Smith and negative for Jones. Conversely, when the numbers differ, Smith pays Jones \)1, reflected by negative values for Smith and positive for Jones.

The payoff matrix makes it possible to visualize the consequences of each player's strategies and is the starting point for analyzing the potential outcomes of the game.

Game Theory

The realm of game theory is concerned with decision-making in competitive environments where the outcome for each player depends on the actions of others. It's a mathematical method for determining optimal strategies in games and conflicts.

Nash Equilibrium is a key concept here, which is a situation where, given the strategies of the other players, no player can benefit by unilaterally changing their strategy. In the context of Smith and Jones, the Nash equilibrium in mixed strategies requires each player to randomize their choices in such a way that neither has an incentive to deviate from this randomized strategy, as doing so would not increase their expected payoffs.

Expected Payoffs

The concept of expected payoffs is fundamentally about anticipating the average outcome or 'payoff' when strategies involve randomization. In mixed strategies, players choose their actions according to certain probability distributions. The expected payoff is then the sum of payoffs of all the possible outcomes, each weighted by the probability of its occurrence.

Calculating the Expected Payoff

In the Smith and Jones game, to determine if a Nash equilibrium in mixed strategies exists, we calculate the expected payoffs for randomizing actions and find probabilities that make these payoffs equal, no matter the opponent's choice. This intricate balance ensures that neither player has a reason to stray from their mixed strategy, indicating that a Nash equilibrium has been achieved.