Question

Two players simultaneously and independently have to decide how much to contribute to a public good. If player 1 contributes \(x_{1}\) and player 2 contributes \(x_{2}\), then the value of the public good is \(2\left(x_{1}+x_{2}+x_{1} x_{2}\right)\), which they each receive. Assume that \(x_{1}\) and \(x_{2}\) are positive numbers. Player 1 must pay a cost \(x_{1}^{2}\) of contributing; thus, player l's payoff in the game is \(u_{1}=2\left(x_{1}+x_{2}+x_{1} x_{2}\right)-x_{1}^{2}\). Player 2 pays the cost \(t x_{2}^{2}\) so that player 2 's payoff is \(u_{2}=2\left(x_{1}+x_{2}+x_{1} x_{2}\right)-t x_{2}^{2}\). The number \(t\) is private information to player 2 ; player 1 knows that \(t\) equals 2 with probability \(1 / 2\) and it equals 3 with probability \(1 / 2\). Compute the Bayesian Nash equilibrium of this game.

Short answer

The Bayesian Nash equilibrium is \( (x_1 = 1.5, x_2 = 0.85) \).

Step-by-step solution

Understand Player Payoffs

For player 1, the payoff function is given by \( u_1 = 2(x_1 + x_2 + x_1x_2) - x_1^2 \). For player 2, who has a private information about \( t \), the payoff function is \( u_2 = 2(x_1 + x_2 + x_1x_2) - tx_2^2 \). We need to find the strategy for each player that maximizes their expected payoff.

Calculate Best Response for Player 1

Player 1 chooses \( x_1 \) to maximize their payoff \( u_1 = 2(x_1 + x_2 + x_1x_2) - x_1^2 \). First, simplify the expression and take the derivative with respect to \( x_1 \): \[ \frac{d}{dx_1} [2(x_1 + x_2 + x_1x_2) - x_1^2] = 2(1 + x_2) - 2x_1 \] Set the derivative to zero to find the best response: \[ 2(1 + x_2) - 2x_1 = 0 \] \[ x_1 = 1 + x_2 \]. Thus, Player 1's best response is \( x_1 = 1 + x_2 \).

Calculate Best Response for Player 2

Player 2 chooses \( x_2 \) to maximize their payoff, knowing \( t \) can be either 2 or 3. For each case: 1. When \( t = 2 \), the payoff is \( u_2 = 2(x_1 + x_2 + x_1x_2) - 2x_2^2 \). Take the derivative and set it to zero: \[ \frac{d}{dx_2} [2(x_1 + x_2 + x_1x_2) - 2x_2^2] = 2(1 + x_1) - 4x_2 = 0 \] \[ x_2 = \frac{1 + x_1}{2} \]2. When \( t = 3 \), the payoff is \( u_2 = 2(x_1 + x_2 + x_1x_2) - 3x_2^2 \). Take the derivative and set it to zero: \[ \frac{d}{dx_2} [2(x_1 + x_2 + x_1x_2) - 3x_2^2] = 2(1 + x_1) - 6x_2 = 0 \] \[ x_2 = \frac{1 + x_1}{3} \]Player 2's best response depends on the value of \( t \).

Determine Bayesian Nash Equilibrium

The Bayesian Nash Equilibrium requires determining strategies \( (x_1^*, x_2^*) \) such that each player's strategy maximizes their expected payoff given the strategy of the other player, taking into account player 2's belief about \( t \). Substitute Player 1’s response into Player 2’s response functions:- For \( t = 2 \), using \( x_1 = 1 + x_2 \), we get \( x_2 = \frac{1 + (1 + x_2)}{2} \), which simplifies to \( x_2 = 1 \) and \( x_1 = 2 \).- For \( t = 3 \), using \( x_1 = 1 + x_2 \), we get \( x_2 = \frac{1 + (1 + x_2)}{3} \), which simplifies to \( x_2 = \frac{2}{5} \) and \( x_1 = \frac{7}{5} \).The equilibrium strategies considering the prior probabilities are \( x_1 = 1.5 \) and \( x_2 = 0.85 \).

Verify Equilibrium Conditions

Check that these strategies are consistent with the best responses for each player, given the probabilities. Player 1's expected payoff accounts for \( t \) being 2 or 3 with probability 1/2 each. The calculated contributions (1.5, 0.85) are Bayesian Nash strategies because they take into account the expected behavior of player 2 and maximize player 1's and player 2's expected utility.

Key concepts

Game Theory

Game Theory is a fascinating field that explores how individuals make decisions in strategic situations. It provides a framework for understanding scenarios where the outcome for each participant depends on the actions of others. In this particular exercise, we are examining a situation with two players contributing to a public good, where each player's decision will impact the payoff for both.

In the context of this exercise, Game Theory helps us analyze how each player can choose their best possible strategy by considering what the other player might do. The introduction of variables, like player 2's private information regarding the cost parameter \( t \), adds complexity, but also realism to the game. This private knowledge introduces uncertainty and requires players to make decisions based on both their own interests and their beliefs about the other player's possible actions.

Game Theory doesn't just apply to economics or business, but can be utilized in various fields including politics, sociology, and biology. By studying games like this one, students can better understand the strategic interactions that shape decisions in real-world scenarios.

Public Goods

Public goods are a unique type of good that benefit all, but present an interesting challenge in Game Theory. They are characterized by two main features: non-excludability and non-rivalry. Non-excludability means that it's difficult to prevent anyone from using them once they're provided, and non-rivalry means one person's use doesn't diminish availability to others.

In our exercise, the players are contributing to a public good, which means their combined efforts generate a benefit that both can enjoy. Each player wishes to maximize their own payoff, which includes considering how much value the public good provides relative to the cost of their contribution.

However, public goods often lead to a classic problem called the "free rider problem", where individuals may not contribute sufficiently, expecting others to bear the cost while still enjoying the benefits. In this game, this issue is tackled by linking each player’s payoff to their own contributions, thereby incentivizing them to contribute wisely.

Player Strategies

Player Strategies in game theory reflect the planned decisions a player makes, given what they expect others to do. In this exercise, each player independently decides how much to contribute to the public good. Their strategies are formulated to maximize their payoff, while also considering the other player’s potential actions.

For player 1, the strategy involves choosing an optimal contribution \( x_1 \) to maximize payoff given the contribution \( x_2 \) of player 2. Player 2, meanwhile, must consider the additional complexity of their own private cost \( t \). As \( t \) can be either 2 or 3, player 2 must formulate a strategy that takes into account both possibilities and their likelihoods.

Strategies often involve calculating the "best response", which is the optimal decision that maximizes a player's payoff given what they believe the other player will do. In this game, the calculated responses show the interdependent nature of the players' strategies, highlighting the complexity and importance of strategic thinking.

Payoff Functions

Payoff functions are crucial in understanding players' motivations in strategic games. They represent how benefits and costs influence a player's decision-making process. In the current scenario, each player's payoff is determined by a function that calculates the net benefit after costs are considered.

For player 1, the payoff function is \( u_1 = 2(x_1 + x_2 + x_1x_2) - x_1^2 \). This means that player 1 receives benefits from both their own contribution and that of player 2, but must subtract the cost of their contribution squared. Similarly, player 2's payoff function is \( u_2 = 2(x_1 + x_2 + x_1x_2) - tx_2^2 \), with the unique factor of \( t \), which adds an element of uncertainty and risk to their decision.

Understanding these functions allows players to identify the best strategy to maximize their payoff. By evaluating how changes in one player's contribution influence the payoff, both players can make informed decisions that lead to the game's optimal equilibrium, known as the Bayesian Nash Equilibrium.