Question

4\. This exercise asks you to combine the investment and hold-up issue from this chapter with the "demand" bargaining game explained in Exercise 4 of Chapter 19. Consider an investment and trade game whereby player 1 first must choose an investment level \(x \geq 0\) at a cost of \(x^{2}\). After player l's investment choice, which player 2 observes, the two players negotiate over how to divide the surplus \(x\). Negotiation is modeled by a demand game, in which the players simultaneously and independently make demands \(m_{1}\) and \(m_{2}\). These numbers are required to be between 0 and \(x\). If \(m_{1}+m_{2} \leq x\) (compatible demands, given that the surplus to be divided equals \(x\) ), then player 1 obtains the payoff \(m_{1}-x^{2}\) and player 2 obtains \(m_{2}\). In contrast, if \(m_{1}+m_{2}>x\) (incompatible demands), then player 1 gets \(-x^{2}\) and player 2 gets 0 . Note that player 1 must pay his investment cost even if the surplus is wasted owing to disagreement. (a) Compute the efficient level of investment \(x^{*}\). (b) Show that there is an equilibrium in which player 1 chooses the efficient level of investment. Completely describe the equilibrium strategies. (c) Discuss the nature of the hold-up problem in this example. Offer an interpretation of the equilibrium of part (b) in terms of the parties' bargaining weights.

Short answer

(a) Efficient investment level is \( x^* = \frac{1}{2} \). (b) Equilibrium strategy: \( m_1 = m_2 = \frac{1}{4} \). (c) Hold-up issue is mitigated by ensuring compatible demands.

Step-by-step solution

Determine Efficient Investment Level

To find the efficient investment level \( x^* \), we need to maximize the total payoff, which is \( x - x^2 \). The efficient level of investment corresponds to maximizing this expression with respect to \( x \). Taking the derivative with respect to \( x \) and setting it to zero: \[ \frac{d}{dx}(x - x^2) = 1 - 2x = 0 \] Solving this gives us \( x^* = \frac{1}{2} \).

Verify Equilibrium Strategy for Part (b)

In equilibrium, players choose strategies that maximize their payoff given the strategies of other players. Suppose player 1 invests \( x^* = \frac{1}{2} \). The surplus to divide is \( x = \frac{1}{2} \). Players simultaneously demand \( m_1 \) and \( m_2 \) under the condition that \( m_1 + m_2 \leq x \). Player 1's payoff is \( m_1 - \left(\frac{1}{2}\right)^2 = m_1 - \frac{1}{4} \), and player 2's payoff is \( m_2 \). For equilibrium, let's check if \( m_1 = \frac{1}{4} \) and \( m_2 = \frac{1}{4} \), making \( m_1 + m_2 = \frac{1}{2} \). This results in compatible demands and payoff of \( 0 \) for Player 1 and \( \frac{1}{4} \) for Player 2, matching their bargaining power.

Analyze Hold-up Problem

The hold-up problem occurs when investment levels are suboptimal due to concerns about post-investment bargaining. Here, even though there is an efficient investment strategy, player 1 underinvests in fear of not obtaining sufficient future payoff (\(-x^2\) if demands are incompatible). However, if both players can commit to optimal equilibrium strategies, player 1 can be assured that \( m_1 + m_2 = x \) will yield investment \( x^* \) with payoff assurance post-investment. The equilibrium reflects player assignments of bargaining weights ensuring \( m_1 = m_2 = \frac{1}{4} \) provide mutual optimality.

Key concepts

Investment Levels

In a bargaining game involving investment, determining the optimal level of investment is crucial for maximizing overall returns. Investment levels refer to the amount player 1 decides to invest, which impacts the surplus that will be divided between the players. Here, player 1 incurs an investment cost of \( x^2 \) and must carefully choose \( x \) to balance the benefits against the costs. The efficient investment level, \( x^* \), is derived by maximizing the net surplus, which is calculated as \( x - x^2 \). To find this optimal point, we take the derivative \( \frac{d}{dx}(x - x^2) = 1 - 2x \), and set it to zero to solve for \( x^* \). This calculation yields \( x^* = \frac{1}{2} \), illustrating that investment should be set at half to be most effective, ensuring the maximal payoff for the system without incurring unnecessary costs.

Bargaining Game

Bargaining games involve negotiation between players on how to distribute a jointly created surplus. In this context, player 1 and player 2 must agree on how to split the surplus \( x \) after player 1 has invested. The players make simultaneous demands \( m_1 \) and \( m_2 \), with the requirement \( m_1 + m_2 \leq x \) for agreements to be compatible.
For successful bargaining, both players need to manage their demands such that the total surplus is not exceeded. If their demands are incompatible and exceed \( x \), the surplus is lost, and player 1 still bears the cost of investment. Negotiations in a bargaining game can reflect the players' strategies and their understanding of each other's bargaining power, which are vital for achieving a mutually beneficial outcome.

Equilibrium Strategy

Equilibrium strategy in game theory represents a situation where each player's strategy maximizes their payoff, given the other player's strategy. In the context of this bargaining game, player 1 must choose an investment \( x^* = \frac{1}{2} \) that allows for the efficient sharing of surplus. The equilibrium occurs when both players' demands, \( m_1 = \frac{1}{4} \) and \( m_2 = \frac{1}{4} \), sum up to \( x \), reflecting an optimal strategy for both parties.
By meeting this equilibrium condition, the demands are compatible, ensuring player 1 receives net zero payoff (after investment costs), while player 2 gains \( \frac{1}{4} \). The strategy here ensures fairness and reflects the cooperative spirit of the game, where each player acknowledges the equitable distribution based on bargaining strengths.

Hold-up Problem

The hold-up problem in game theory arises when one party is hesitant to invest optimally due to fears of future exploitation in bargaining situations. In this exercise, player 1 might underinvest due to concerns about not receiving appropriate compensation after the investment is made. Understanding this issue is crucial since it complicates the decision to invest efficiently.
The equilibrium strategy helps mitigate the hold-up problem by ensuring appropriate bargaining outcomes, where mutually optimal demands \( m_1 + m_2 = x \) provide assurances to player 1. Commitments to established equilibrium strategies allow player 1 to confidently invest \( x^* \), knowing the return will be fair and profitable post-bargaining. This balance of power and pre-negotiated outcomes helps reduce the risk of suboptimal investment due to fear of exploitation, supporting a healthy investment climate.