Question

Persons 1 and 2 are forming a firm. The value of their relationship depends on the effort that they each expend. Suppose that person $i$ 's utility from the relationship is $x_j^2+x_j-x_i{x}_j$, where $x_i\ge 0$ is person $i$ 's effort and $x_j$ is the effort of the other person $(i=1,2)$. (a) Compute the partners' best-response functions and find the Nash equilibrium of this game. Is the Nash equilibrium efficient? (b) Now suppose that the partners interact over time, which we model with the infinitely repeated version of the game. Let $\delta$ denote the discount factor of the players. Under what conditions can the partners sustain some positive effort level $x=x_1=x_2$ over time? (Postulate strategies and then derive a condition under which the strategies form an equilibrium in the repeated game.) (c) Comment on how the maximum sustainable effort depends on the partners' patience.

Short answer

(a) Nash equilibrium is $(0,0)$ and is inefficient. (b) Effort is sustainable if $\delta >1-x$. (c) Maximum sustainable effort rises with patience.

Step-by-step solution

Determine Best-Response Functions

First, we need to find the best response functions for each player. A player's utility is given as $x_j^2+x_j-x_i{x}_j$. To find the best-response function of player $i$, we differentiate this utility with respect to $x_i$ and set the derivative to zero.$$\frac{d}{d{x}_i}(x_j^2+x_j-x_i{x}_j)=-x_j=0$$This tells us $x_j=0$, implying the best response of each player is that the other player should exert no effort. Thus, the best-response functions are:$$x_1^{\ast }=0,x_2^{\ast }=0$$

Find Nash Equilibrium

The Nash Equilibrium occurs when each player's strategy is the best response to the other player's strategy. From the best-response functions:$$x_1^{\ast }=0,x_2^{\ast }=0$$This means the Nash Equilibrium is $(x_1,x_2)=(0,0)$. In this equilibrium, neither player exerts effort since no deviation can yield better utility.

Analyze Efficiency of Nash Equilibrium

To check efficiency, consider the joint utility of both players:PLAYER 1 UTILITY: $x_2^2+x_2$PLAYER 2 UTILITY: $x_1^2+x_1$Adding these, joint utility is maximized with effort since both utilities increase with positive $x_1$ and $x_2$. As $(0,0)$ is not maximizing joint utility, Nash equilibrium is not efficient.

Sustaining Effort in Infinitely Repeated Game

In the repeated game, define a strategy where players exert positive effort $x$ if previous rounds have shown full effort, and $0$ if not. For positive effort to be sustained, the discounted future payoffs must outweigh short-term gains from defecting.Total utility if sustaining effort: $U_{extcoop}=\frac{x^2}{1-\delta }$.Utility from defecting once: $U_{extdefect}=x$ immediately, but zero thereafter.For $x:U_{\text{coop}}>U_{\text{defect}}$,$$\frac{x^2}{1-\delta }>x$$Simplifying, $x>(1-\delta )$. Thus $\delta >1-x$ is necessary for sustainable effort.

Impact of Patience on Maximum Sustainable Effort

As $\delta$, or patience, increases, $1-\delta$ approaches zero, allowing for higher $x$ in sustainable cooperation. This means the more patient the partners (higher $\delta$), the higher the effort level $x$ they can maintain over time. Thus, maximum sustainable effort increases with partner patience.

Key concepts

Nash Equilibrium

In game theory, the term 'Nash Equilibrium' refers to a situation where no player can benefit by changing their strategy while the other player's strategies remain unchanged. Let's imagine two individuals forming a firm where each person's utility depends on their effort levels and the efforts of the other person. This creates an interdependent scenario with strategic decision-making.

In the provided exercise, the Nash equilibrium is reached when both individuals decide to exert zero effort. This means there's no incentive for either party to increase effort unilaterally, since doing so doesn't improve their utility under the current response of the other party. However, this equilibrium is not efficient because it doesn't maximize the combined utility of both individuals, which means there might be potential for improvement if they cooperate rather than sticking to limited selfish strategies.

Best Response Functions

Finding the best response functions is about determining each player's optimal strategy in response to the other player's actions. It involves calculating how one player's utility changes with their efforts, given the effort of the other player.

In our exercise scenario, the best response function for each person is identified by finding where the derivative of their utility function, in terms of their own effort, equals zero. This implies they make the best decision to maximize their utility. For both individuals in the firm, the best response is for the other to expend zero effort, leading to these best-response functions:

• Player 1's best response: exert no effort as Player 2's effort doesn't improve their utility.
• Player 2's best response: similarly, exert no effort given Player 1's effort choice.

This deduction results in both players pulling back effort, leading up to the Nash Equilibrium at zero effort levels.

Repeated Games

Repeated games consider a situation where players encounter the same game multiple times. The decision-making extends over these multiple periods and can significantly alter strategies compared to a one-time interaction.

For this firm's formation exercise, envision both people repeatedly playing the same game across infinite rounds. With each round, there's an opportunity for changing strategies based on past outcomes. The primary goal in these settings is often about sustaining cooperation over time, expressing repeated interactions in terms of a discount factor, edged by previous experiences.
Players might start by agreeing to exert positive effort, but if one reduces their effort, the other could switch strategies as a form of punishment in future rounds. Such a punishment mentality in repeated games pushes players to stay cooperative throughout, provided the future gains overwhelmingly offset any instant benefits from shirking effort.

Discount Factor

The discount factor ($\delta$) is crucial in repeated games, signifying how much future payoffs are valued compared to present payoffs. It reflects the patience level of the players in the game.

In our scenario, if both players place high value on future payoffs (i.e., they are patient), they could sustain a higher level of effort over time. This is because as $\delta$ approaches 1, the individuals are more inclined to cooperate to maintain high future benefits.

The logic here is straightforward: patience, illustrated through a near-unity discount factor, incentivizes maintaining cooperative strategies, as the long-term benefits seem more appealing than short-term gains from defection. Therefore, sustaining collective effort largely hinges on how much $\delta$ exceeds the threshold necessary to support collaboration, $1-x$, dictating the persistence of effort levels over time.