Question

Suppose we have a game where \(S_{1}=\\{\mathrm{H}, \mathrm{L}\\}\) and \(S_{2}=\\{\mathrm{X}, \mathrm{Y}\\}\). If player 1 plays \(\mathrm{H}\), then her payoff is \(z\) regardless of player 2 's choice of strategy; player 1 's other payoff numbers are \(u_{1}(\mathrm{~L}, \mathrm{X})=0\) and \(u_{1}(\mathrm{~L}, \mathrm{Y})=10 .\) You may choose any payoff numbers you like for player 2 because we will only be concerned with player l's payoff. (a) Draw the normal form of this game. (b) If player 1's belief is \(\theta_{2}=(1 / 2,1 / 2)\), what is player 1's expected payoff of playing H? What is his expected payoff of playing L? For what value of \(z\) is player 1 indifferent between playing \(\mathrm{H}\) and \(\mathrm{L}\) ? (c) Suppose \(\theta_{2}=(1 / 3,2 / 3)\). Find player l's expected payoff of playing \(\mathrm{L}\).

Short answer

(a) See step 2. (b) Expected payoffs: H = z, L = 5; Indifference if z=5. (c) L's payoff = 6.67.

Step-by-step solution

Define Player 1's Payoffs

We have that Player 1's payoffs are as follows: - For strategy set \((H, X)\), payoff is \(z\).- For strategy set \((H, Y)\), payoff is \(z\).- For strategy set \((L, X)\), payoff is 0.- For strategy set \((L, Y)\), payoff is 10.

Draw the Normal Form

The normal form representation of the game for Player 1, where the rows represent Player 1's strategies and the columns represent Player 2's strategies, is:\[\begin{array}{c|c|c} & X & Y \\hlineH & z & z \L & 0 & 10 \\end{array}\]

Calculate Expected Payoff for H (Belief \(\theta_{2}=(1/2,1/2)\))

Given Player 1's belief \(\theta_{2}=(1/2,1/2)\), the expected payoff for playing \(H\) is:\[E(H) = \theta_{2}(X)\cdot u_{1}(H, X) + \theta_{2}(Y)\cdot u_{1}(H, Y) = \frac{1}{2}\cdot z + \frac{1}{2}\cdot z = z\]

Calculate Expected Payoff for L (Belief \(\theta_{2}=(1/2,1/2)\))

The expected payoff for playing \(L\) is:\[E(L) = \theta_{2}(X)\cdot u_{1}(L, X) + \theta_{2}(Y)\cdot u_{1}(L, Y) = \frac{1}{2}\cdot 0 + \frac{1}{2}\cdot 10 = 5\]

Find Indifference Condition for Player 1 between H and L

To find when Player 1 is indifferent between \(H\) and \(L\), set their expected payoffs equal:\[z = 5\]Player 1 is indifferent if \(z = 5\).

Calculate Expected Payoff for L (Belief \(\theta_{2}=(1/3,2/3)\))

With belief \(\theta_{2}=(1/3,2/3)\), calculate the expected payoff for playing \(L\):\[E(L) = \frac{1}{3}\cdot 0 + \frac{2}{3}\cdot 10 = \frac{20}{3} \approx 6.67\]

Key concepts

Normal Form Representation

In game theory, the normal form of a game is a matrix that represents the strategic possibilities available to players and their respective payoffs. It allows us to visualize the game and understand how players' choices interact. In this example:

• Player 1 has two strategies: H and L.
• Player 2 likewise can pick between X and Y.

By organizing the players’ strategies into a grid, we can easily see the potential outcomes and payoff values.
The payoff matrix for Player 1 from our game setup looks like this:\[\begin{array}{c|c|c} & X & Y \\hlineH & z & z \L & 0 & 10 \\end{array}\]In this matrix:

• The first row corresponds to Player 1 playing H, which results in a payoff of \(z\), whether Player 2 chooses X or Y.
• The second row depicts Player 1 playing L, where the payoff is \(0\) if Player 2 chooses X and \(10\) if Player 2 selects Y.

This representation is crucial for analyzing the strategies and their respective outcomes.

Expected Payoff Calculation

Calculating the expected payoff is essential in determining what a rational player would likely choose when faced with uncertainty about the other player's strategies. This involves weighing each possible payoff by the probability of the corresponding strategy being selected by the opponent.
To calculate the expected payoff for Player 1 when playing H, given Player 2's belief of choosing each of their available strategies with equal probabilities, \(\theta_{2}=(1/2,1/2)\):\[E(H) = \theta_{2}(X) \cdot u_{1}(H, X) + \theta_{2}(Y) \cdot u_{1}(H, Y) = \frac{1}{2} \cdot z + \frac{1}{2} \cdot z = z\]Similarly, we calculate the expected payoff for Player 1 playing L:\[E(L) = \theta_{2}(X) \cdot u_{1}(L, X) + \theta_{2}(Y) \cdot u_{1}(L, Y) = \frac{1}{2} \cdot 0 + \frac{1}{2} \cdot 10 = 5\]Here, the expected payoffs are derived using the assigned probabilities and the resulting payoffs from Player 1's actions.

Indifference Condition

An indifference condition occurs when a player has no preference between two strategies because they yield the same expected payoff. This is particularly important in mixed strategy games where strategies might not be pure but instead are chosen according to a probability distribution.
To find the value of \(z\) for which Player 1 is indifferent between choosing H or L, we equate the expected payoffs:\[z = 5\]This equation tells us that when the payoff of choosing H, \(z\), equals 5, Player 1 becomes indifferent. Both strategies provide an equal expected payoff in the eyes of Player 1 under the given beliefs. Being indifferent implies that Player 1 might be equally likely to opt for either H or L as either satisfies their expected payoff requirements.

Mixed Strategy

A mixed strategy in game theory involves a player randomizing over two or more strategies, each with a specific probability. It is particularly relevant in situations where no pure strategy provides a definitive advantage or when an indifference condition applies.
Even when facing probabilities that might lead the player to favor different strategies based on altered expectations, mixed strategies maintain balance. For instance, suppose Player 1 faces new probabilities, such as \(\theta_{2}=(1/3,2/3)\) for Player 2's choices; we performed the recalculation for L as:\[E(L) = \frac{1}{3} \cdot 0 + \frac{2}{3} \cdot 10 = \frac{20}{3} \approx 6.67\]In the context of mixed strategies, this might lead Player 1 to reconsider the probability distributions of their strategies, potentially choosing to randomize between H and L according to the new expected payoffs.
This approach allows players to remain unpredictable, which can be strategic in many competitive scenarios. Mixed strategies thus extend the strategic landscape beyond deterministic gameplay.