Question

Consider the following two-player team production problem. The players simultaneously choose effort levels ( \(a_{1}\) for player 1 and \(a_{2}\) for player 2 ), and revenue is given by \(2 k\left[a_{1}+a_{2}\right]\). Suppose that player \(i\) 's cost of effort is \(a_{i}^{2}\) in monetary terms. Revenue is split equally between the players, so player \(i\) 's payoff is \(k\left[a_{1}+a_{2}\right]-a_{i}^{2}\). Assume that the value of \(k\) is privately observed by player 1 prior to her selection of \(a_{1}\). Player 2 does not know the value of \(k\) before choosing \(a_{2}\). Player 2 knows only that \(k\) is either 4 or 8 , with these being equally likely. (a) Find the Bayesian Nash equilibrium (effort levels) for this game. (b) Consider a variation of the game in which, after observing \(k\) and prior to effort selection, player 1 can provide evidence of the value of \(k\) to player 2. That is, after either realization of \(k\), player 1 chooses either \(\mathrm{E}\) or \(\mathrm{N}\). If she selects \(\mathrm{E}\), then \(k\) is revealed to player 2 ; otherwise, player 2 observes nothing other than that player 1 selected \(\mathrm{N}\). Represent this game in the extensive form and calculate the perfect Bayesian equilibria. Are there any perfect Bayesian equilibria in which, at the time player 2 selects \(a_{2}\), she is uncertain of \(k\) ? (c) Would player 1 prefer to have the option of revealing \(k\), as in part (b) relative to part (a)? Explain.

Short answer

(a) Bayesian Nash Equilibrium: \(a_1=k/2\), \(a_2=1.5\). (b) PBE allows uncertainty if \(E\) and \(N\) are strategies. (c) Yes, revealing \(k\) can benefit Player 1.

Step-by-step solution

Understanding Player 1's Strategy

Player 1 observes the value of \(k\) and must choose effort \(a_1\). The payoff function for Player 1 is given by \( k[a_1 + a_2] - a_1^2 \). Thus, Player 1 will maximize her payoff by solving the optimization problem \( \max_{a_1} \ k[a_1 + a_2] - a_1^2 \). The first-order condition for maximization gives \( k - 2a_1 = 0 \), solving this gives \( a_1^* = \frac{k}{2} \). Hence, Player 1's best response to any \(a_2\) is \(a_1^* = \frac{k}{2} \).

Determining Player 2's Expected Payoff

Player 2 knows \(k\) can be either 4 or 8, each with probability 0.5, and does not observe it prior to choosing \(a_2\). The expected payoff for Player 2 is therefore \(\frac{1}{2}[4(a_1 + a_2) - a_2^2] + \frac{1}{2}[8(a_1 + a_2) - a_2^2]\). Incorporating the expected values of \(k\), substitute \( a_1 = a_1^* = \frac{4}{2} = 2 \) or \(a_1 = 4\), resulting in the expected payoff as a function of \(a_2\).

Calculating Player 2's Best Response

Taking the derivative of Player 2’s expected payoff with respect to \(a_2\), set the derivative equal to zero to find the optimal \(a_2\). The expected payoff function reduces to \((3a_2 + 6) - a_2^2\). The first order condition \(3 - 2a_2 = 0\) gives \(a_2^* = 1.5\). Thus, Player 2's best response is \(a_2^* = 1.5\).

Bayesian Nash Equilibrium

Since Player 1 will play \(a_1 = \frac{k}{2}\), and Player 2 plays \(a_2 = 1.5\), this strategy pair constitutes the Bayesian Nash Equilibrium. Specifically, \(a_1 = 2\) when \(k = 4\), \(a_1 = 4\) when \(k = 8\), and \(a_2 = 1.5\) regardless of \(k\).

Extensive Form of the Modified Game

Now, consider the variation where Player 1 can choose to reveal \(k\) (E) or not (N) to Player 2 before choosing \(a_1\). This game can be represented by a tree where Player 1 first chooses between E and N, then based on that, Player 2 observes either \(k\) or nothing, and selects \(a_2\).

Strategies and Perfect Bayesian Equilibria in Modified Game

For any equilibrium, we check whether Player 2 must be uncertain of \(k\) when choosing \(a_2\). If Player 1 reveals \(k\) by choosing \(E\), Player 2's choice is based on actual \(k\). If Player 1 chooses \(N\), Player 2's expectation remains uncertain. Player 1 can manipulate the strategy by switching between revealing and not revealing \(k\) in different equilibria strategies.

Preference of Player 1 for Revealing Option

Player 1 can benefit from the new strategy (E or N), depending on how it affects Player 2's actions relative to her information. If revealing \(k\) leads to Player 2 choosing a more cooperative \(a_2\), Player 1 may prefer this option over the original scenario where Player 2 is uncertain. Thus, Player 1 likely prefers the option to reveal \(k\).

Key concepts

Game Theory

Game theory is an essential field of applied mathematics that explores strategic interactions among rational decision-makers. In the context of the exercise provided, game theory helps understand how each player in the team production problem makes decisions based on payoffs that depend not only on their actions but also on the actions of others. The goal is to find the strategies leading to the most beneficial outcomes for each player.

A fundamental part of game theory is to analyze situations in which players simultaneously choose strategies, as seen in this two-player scenario. Players make decisions without knowing the choices of their counterparts.

• Simultaneous games: Decisions are made by players at the same time, without knowledge of the other player's strategy.
• Payoff functions: Reflect the benefits or costs resulting from the combination of chosen strategies.
• Players' objectives: Typically revolve around maximizing their respective payoffs through optimal strategy selection.

These elements serve as the foundation upon which more advanced concepts, such as Bayesian Nash Equilibrium, are built.

Perfect Bayesian Equilibrium

Perfect Bayesian Equilibrium (PBE) refines the traditional Nash Equilibrium concept by incorporating aspects of belief systems and incomplete information. It is especially useful in dynamic games where players face uncertainties, like in the modified version of the team production problem.

In PBE, each player forms beliefs about the unknown aspects of the game, which informs their decision-making process. These beliefs are updated via Bayesian principles as new information comes to light.

• Belief updating: As players receive information, their beliefs about unknown game elements change.
• Consistency: Player strategies must be consistent with the updated beliefs, ensuring optimal decisions based on available information.
• Dynamic adjustments: Players adjust strategies dynamically as new evidence emerges, like Player 1 potentially revealing the value of \(k\).

PBE thus helps predict player behavior in more complex settings, adding depth to the analysis beyond static Nash Equilibrium.

Player Strategies

Player strategies in game theory refer to the specific choices or actions a player decides upon within a game setting. In the team production problem, each player's strategy involves deciding their effort level to maximize individual payoffs.

Player 1, who knows the private information about \(k\), selects effort based on optimizing her payoff function, while Player 2 must strategically operate under uncertainty, calculating expected payoffs.

• Player 1's Strategy: Due to the private knowledge of \(k\), her strategy directly utilizes this information to optimize her effort, \(a_1\), ensuring her best response is \(a_1^* = \frac{k}{2}\).
• Player 2's Strategy: Faces uncertainty about \(k\) and hence bases efforts \(a_2\) on probable outcomes, achieving a best response at \(a_2^* = 1.5\).
• Strategic interaction: Strategies depend not just on individual pay-off but are deeply intertwined with anticipation of the other player's actions.

Employing optimal player strategies helps illuminate characteristics of equilibrium states, like the Bayesian Nash Equilibrium here.

Team Production Problem

The team production problem illustrates a classic economic conflict where individual motivations might not align with overall team goals, or the global optimum. This challenge is underscored by issues of effort exertion linked to combined production outcomes.

In this problem, each player's effort directly impacts the team's revenue. However, it also incurs a personal cost in terms of effort exertion, outlined as \(a_{i}^2\). The crux of the problem is balancing effort contribution against individual payoff derived from the collective output.

• Shared revenue: The total revenue produced is a function of combined efforts and is split equally among players, influencing their strategic decisions.
• Personal costs: The cost incurred by individual efforts, measured in monetary terms, dictates how much effort is justifiable for maximizing personal gains.
• Collaboration vs. competition: While the goal is team success, individual payoff maximization can lead to differing effort strategies.

Understanding the team's production problem is vital for analyzing cooperative behaviors and strategizing to align individual objectives with collective outcomes.