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Chapter 11: Problem 9

Prove that every \(2 \times 2\) game has a Nash equilibrium (in either pure or mixed strategies). Do this by considering the following general game and breaking the analysis into two categories: (a) one of the pure-strategy profiles is a Nash equilibrium, and (b) none of the pure-strategy profiles is a Nash equilibrium.

Short Answer

Expert verified
Every \(2 \times 2\) game has a Nash equilibrium in either pure or mixed strategies.

Step by step solution

01

Understanding Pure Strategies

In a general \(2 \times 2\) game, we have two players: Player 1 chooses between strategies \(A\) and \(B\), while Player 2 chooses between \(C\) and \(D\). The payoff matrix for Player 1 is given by \([a, b]\) for strategies \(A\) and \(B\) chosen by Player 2 and \([c, d]\) for \(A\) and \(B\) chosen by Player 1. Similarly, Player 2's payoffs are represented by \([e, f]\) and \([g, h]\).
02

Identifying Nash Equilibria in Pure Strategies

Check each pure-strategy profile: \((A,C), (A,D), (B,C), (B,D)\) for Nash equilibria. A profile is a Nash equilibrium if neither player can benefit by unilaterally changing their strategy. For example, for the profile \((A,C)\) to be a Nash equilibrium, Player 1's payoff \(a\) should be greater than or equal to \(c\) given \(C\), and Player 2's payoff \(e\) should be greater than or equal to \(f\) given \(A\). Similar checks are performed for other profiles.
03

Analyzing Absence of Pure Strategy Nash Equilibrium

Assume none of the four strategy profiles are Nash equilibria in pure strategies, meaning for each profile, a player can benefit from changing strategies. This indicates the need to explore mixed strategies, where players randomize over their two strategies.
04

Constructing Mixed Strategy Nash Equilibrium

In a mixed strategy, Player 1 chooses \(A\) with probability \(p\) and \(B\) with probability \(1-p\). Player 2 chooses \(C\) with probability \(q\) and \(D\) with probability \(1-q\). The gains from each strategy need to be equal for a player to be indifferent, so we solve the systems of equations: \(pe + (1-p)g = pf + (1-p)h\) for Player 2, and \(qa + (1-q)c = qb + (1-q)d\) for Player 1, to find \(p\) and \(q\).
05

Proof Completion

Solving the above equations gives values for \(p\) and \(q\), ensuring each player's expected payoff from their strategies is equal, indicating a Nash equilibrium in mixed strategies. Thus, even if no pure strategy equilibrium exists, a mixed strategy equilibrium is guaranteed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mixed Strategies
In game theory, mixed strategies involve players randomizing over multiple strategies rather than sticking to a single, deterministic strategy. This approach is particularly useful when pure strategies do not yield a Nash Equilibrium (NE). In essence, a mixed strategy assigns a probability to each available action, allowing for a more flexible and unpredictable approach.

When applying mixed strategies in a game, players are often able to create a situation where no opponent can benefit from changing their strategy, given the mixed strategy adopted by the other player. This is particularly important in two-player games where no pure strategy NE is present.

To find a mixed strategy NE, we set conditions where players are indifferent to their available choices. Typically, system of equations are used to balance expected payoffs. This ensures that any shift in strategy does not yield any additional expected gain.
  • This requires calculating expected payoffs for each strategic choice.
  • Indifference conditions help find the exact probabilities to randomize over strategies.
Working through a mixed strategy analysis can seem complex, but it offers a mathematically sound way to find equilibria in games without obvious solutions.
Pure Strategies
Pure strategies mean selecting one particular strategy without any randomness involved. Each strategy combination in a game forms what is known as a pure strategy profile. In the context of Nash Equilibrium, a pure strategy NE occurs when no player can benefit from unilaterally changing their chosen strategy.

In a simple \(2 \times 2\) game, there are typically four possible pure strategy profiles for players to consider: \( (A, C), (A, D), (B, C)\), and \( (B, D)\). When checking for a pure strategy NE, it involves verifying if maintaining one's strategy choice offers the highest payoff, given the opponent's choice.
  • This requires evaluating all possible strategy combinations.
  • Ensure each chosen strategy yields the best response payoff.
If any single profile satisfies these conditions, it indicates the presence of a pure strategy Nash Equilibrium. Otherwise, it suggests a need to explore mixed strategies.
Game Theory
Game theory is the study of strategic decision-making where multiple players interact, considering each other's potential decisions. It's a framework for understanding how rational players reach decisions in situations where their outcomes depend on the actions of others.

Within game theory, players seek to optimize their payoffs. A key feature of these games is the Nash Equilibrium, which represents conditions under which players have chosen best responses to one another's strategies, resulting in no incentive to deviate unilaterally.
  • Game theory is often applied in economics, politics, and negotiation scenarios.
  • It forms the foundation for analyzing competitive and cooperative behaviors.
This theory incorporates concepts like pure and mixed strategies to predict and analyze outcomes in complex strategic situations. It allows for the resolution of dilemmas and strategic interactions that arise in competitive settings.
Payoff Matrix
A payoff matrix is a crucial tool in game theory used to capture the interplay of decisions by the players in a game. It consists of a grid that showcases the outcomes (or payoffs) of all possible strategy combinations between the players.

In a \(2 \times 2\) game, the matrix provides a clear visualization of each player's choices and respective payoffs for every possible scenario. Each cell of the matrix corresponds to a strategic choice of both players, showing their respective payoffs.
  • Row and column intersections denote strategic interactions.
  • The matrix helps identify Nash Equilibria through evaluation of best responses.
Understanding and constructing payoff matrices helps break down complex strategic games into manageable sections, facilitating the analysis needed to determine optimal strategies.

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Most popular questions from this chapter

Consider a variant of the Bertrand game with capacity constraints that was analyzed in this chapter. Suppose that firm 1's capacity constraint is \(c_{1}\) and firm 2 's capacity constraint is \(c_{2}\), where \(c_{1}, c_{2} \geq 5\). That is, firm 1 can produce at most \(c_{1}\) units, and firm 2 can produce at most \(c_{2}\) units. As before, if \(p_{1}=p_{2} \leq 1\), then five consumers buy from firm 1 , and five consumers buy from firm 2, as in the standard model. Furthermore, if firm \(i\) charges a lower price than does firm \(j\), so that \(p_{i}c_{2}\), then \(F_{1}\) has a discontinuity at 1 (a mass point for firm 1's mixed strategy). In any case, firm 1's equilibrium expected payoff is \(c_{1} \underline{p}\) and firm 2 's equilibrium expected payoff is \(c_{2} \underline{p}\).

Consider the following three-player team production problem. Simultaneously and independently, each player chooses between exerting effort (E) or not exerting effort ( \(N\) ). Exerting effort imposes a cost of 2 on the player who exerts effort. If two or more of the players exert effort, each player receives a benefit of 4 regardless of whether she herself exerted effort. Otherwise, each player receives zero benefit. The payoff to each player is her realized benefit less the cost of her effort (if she exerted effort). For instance, if player 1 selects \(\mathrm{N}\) and players 2 and 3 both select \(\mathrm{E}\), then the payoff vector is \((4,2,2)\). If player 1 selects \(E\) and players 2 and 3 both select \(N\), then the payoff vector is \((-2,0,0)\). (a) Is there a pure-strategy equilibrium in which all three players exert effort? Explain why or why not. (b) Find a symmetric mixed-strategy Nash equilibrium of this game. Let \(p\) denote the probability that an individual player selects \(\mathrm{N}\).

Consider a game between a police officer (player 3 ) and two drivers (players 1 and 2). Player 1 lives and drives in the Clairemont neighborhood of San Diego, whereas player 2 lives and drives in the Downtown area. On a given day, players 1 and 2 each have to decide whether or not to use their cell phones while driving. They are not friends, so they will not be calling each other. Thus, whether player 1 uses a cell phone is independent of whether player 2 uses a cell phone. Player 3 (the police officer) selects whether to patrol in Clairemont or Downtown. All of these choices are made simultaneously and independently. Note that the strategy spaces are \(S_{1}=\\{\mathrm{U}, \mathrm{N}\\}, S_{2}=\\{\mathrm{U}, \mathrm{N}\\}\), and \(S_{3}=\\{\mathrm{C}, \mathrm{D}\\}\), where "U" stands for "use cell phone," " \(\mathrm{N}\) " means "not use cell phone," "C"' stands for "Clairemont," and "D" means "Downtown." Suppose that using a cell phone while driving is illegal. Furthermore, if a driver uses a cell phone and player 3 patrols in his or her area (Clairemont for player 1, Downtown for player 2), then this driver is caught and punished. A driver will not be caught if player 3 patrols in the other neighborhood. A driver who does not use a cell phone gets a payoff of zero. A driver who uses a cell phone and is not caught obtains a payoff of 2 . Finally, a driver who uses a cell phone and is caught gets a payoff of \(-y\), where \(y>0\). Player 3 gets a payoff of 1 if she catches a driver using a cell phone, and she gets zero otherwise. (a) Does this game have a pure-strategy Nash equilibrium? If so, describe it. If not, explain why. (b) Suppose that \(y=1\). Calculate and describe a mixed-strategy equilibrium of this game. Explain whether people obey the law. (c) Suppose that \(y=3\). Calculate and describe a mixed-strategy equilibrium of this game. Explain whether people obey the law.

Consider a game with \(n\) players. Simultaneously and independently, the players choose between \(\mathrm{X}\) and \(\mathrm{Y}\). That is, the strategy space for each player \(i\) is \(S_{i}=\\{\mathrm{X}, \mathrm{Y}\\}\). The payoff of each player who selects \(\mathrm{X}\) is \(2 m_{x}-m_{x}^{2}+3\), where \(m_{x}\) is the number of players who choose X. The payoff of each player who selects \(\mathrm{Y}\) is \(4-m_{y}\), where \(m_{y}\) is the number of players who choose \(\mathrm{Y}\). Note that \(m_{x}+m_{y}=n\). (a) For the case of \(n=2\), represent this game in the normal form and find the pure-strategy Nash equilibria (if any). (b) Suppose that \(n=3\). How many Nash equilibria does this game have? (Note: you are looking for pure-strategy equilibria here.) If your answer is more than zero, describe a Nash equilibrium. (c) Continue to assume that \(n=3\). Determine whether this game has a symmetric mixed-strategy Nash equilibrium in which each player selects \(\mathrm{X}\) with probability \(p\). If you can find such an equilibrium, what is \(p\) ?

Player 1 (the "hider") and player 2 (the "seeker") play the following game. There are four boxes with lids, arranged in a straight line. For convenience, the boxes are labeled A, B, C, and D. The administrator of the game gives player 1 a \(\$ 100\) bill, and player 1 must hide it in one of the four boxes. Player 2 does not observe where player 1 hides the \(\$ 100\) bill. Once player 1 has hidden the bill, player 2 must open one (and only one) of the boxes. If the money is in the box that player 2 opens, then player 2 keeps the \(\$ 100\). If it is not, player 1 gets to keep the \(\$ 100\). (a) Does this game have a pure-strategy Nash equilibrium? (b) Find the mixed-strategy Nash equilibrium. (c) Suppose it is common knowledge that player 1 likes the letter "A" and would get extra satisfaction from putting the money in box A. Let this satisfaction be equivalent to receiving \(\$ 20\). Assume this is in addition to any money received in the game. How does player 1's preference for the letter "A" affect the equilibrium mixing probabilities? Calculate the new equilibrium strategy profile if you can. (d) Describe the equilibria of the game in which player 1's extra satisfaction from selecting box \(\mathrm{A}\) is equivalent to receiving \(\$ 120\).
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