Question

8\. Here is a description of interaction between two players who are considering a possible business partnership. First the players simultaneously choose whether to make an investment. Investment entails a personal \(\cos t\) of 3 ; not investing costs nothing. The investment choices become common knowledge. Then the players jointly decide whether to form a partnership firm and, if so, how to divide the profit from the firm. If both players invested, then the firm's profit is 16 . If exactly one player invested or if neither invested, then the firm's profit is 12 . If the players decide not to form the firm, then each player \(i\) gets a default payoff of \(x-3\) if player \(i\) invested and zero if player \(i\) did not invest. The default payoff of \(x-3\) includes the cost of investment plus some value \(x\) that represents what player \(i\) can obtain by using his investment in other endeavors. Assume that the players divide surplus according to the standard bargaining solution with equal bargaining weights. (a) What outcome maximizes the joint value? That is, what are the efficient investment choices? (b) Describe conditions on \(x\) such that there is a negotiation equilibrium in which both players invest. Show that this is an equilibrium. (c) In light of your answers to parts (a) and (b), briefly provide some intuition for your answers in relation to the "hold-up" problem.

Short answer

(a) Both players invest. (b) If \(x > 9\). (c) Hold-up mitigated by ensuring investment recovery.

Step-by-step solution

Determine Joint Value Maximizing Outcome

To maximize the joint value, both players should make decisions that result in the highest combined payoff. If both invest, the firm earns a profit of 16, and the cost is 3 for each, leading to a net joint payoff of 16 - 6 = 10. If one or neither invests, the profit is 12, with no investment cost, resulting in a net payoff of 12. Therefore, both investing maximizes joint value.

Analyze Payoff Structure for Negotiation Equilibrium

We need to determine conditions on \(x\) for both players to have an incentive to invest. If both invest, they can form a firm and earn 16 to be divided equally minus costs (net gain is \(8 - 3 = 5\) each). If one player chooses not to invest, that player still gains \(6\) from negotiating the profit \(\frac{12}{2} = 6\) without the investment cost. For each to invest, the net benefit must exceed this default gain: \(x - 3 > 6\). Thus \(x > 9\).

Tax to Support Equilibrium Strategy

To confirm this is a negotiation equilibrium, assume the condition \(x > 9\). If a player unilaterally deviates and does not invest, their payoff is \(6\). Given \(x > 9\), the payoff from investing \(x - 3\) is better than \(6\). Each player's dominant strategy is to invest if others' actions don't change responses best for individual gain.

Provide Intuition Relating to Hold-Up Problem

The hold-up problem occurs when parties under-invest in relation to potential joint gains through fear of not recovering investments from a negotiated outcome. Here, \(x > 9\) ensures players recover their investment and realize surplus, mitigating hold-up issues by aligning individual payoff with mutual investment, securing agreements at optimum cost-benefit.

Key concepts

Negotiation Equilibrium

In game theory, particularly when exploring strategic interactions like partnerships, negotiation equilibrium is a cornerstone concept. This equilibrium ensures that all participants in a negotiation are making decisions that keep the outcomes optimal for everyone involved. In the business partnership scenario, players need to determine whether or not to invest with the knowledge that future profits will be negotiated. The negotiation equilibrium comes into play when both players agree to invest, understanding that their profits will be divided equally following a joint decision on partnership formation.
The key condition here is to make sure that each player’s investment decision provides a payoff higher than any alternative option available if they choose not to invest. As illustrated, this establishes the condition on the return value of investment, denoted by "x." Specifically, players will find it beneficial to invest when their potential gross payoff from the invested partnership (greater than 9, considering costs) is higher than the default scenario where they earn 6 without investment. Thus, negotiation equilibrium hinges on incentives that compel mutual investment by making it the best strategic choice.

Joint Value Maximization

Joint value maximization is an optimization strategy where the combined gains of all participants in a venture are prioritized. It seeks to ensure maximum possible returns through cooperative strategies rather than individualistic gains. In our business exercise, joint value maximization is achieved when both players decide to invest, producing a joint profit of 16. Despite the investment costs of 3 for each player, the overall net payoff exceeds scenarios where only one or neither player invests.
This concept becomes vital in scenarios where partnership formation depends on balancing investment costs against potential collective profits. Here, players must realize that their collective investment leads to overall greater returns, allowing them to share a surplus that exceeds any individual withholding. In essence, joint value maximization advocates for strategies that promote equity and shared benefits in decision-making processes, which results in achieving the highest sum benefit feasible for the group.

Hold-Up Problem

The hold-up problem in economic theory highlights a situation where potential partners hesitate to make initial investments due to fears of not recovering their costs from the joint endeavor's proceeds. This reluctance often leads to underinvestment, affecting the joint efficiency of the partnerships.
In the given exercise, the hold-up problem is mitigated by ensuring "x" exceeds 9. This threshold guarantees that the invested players anticipate recovering their costs and earning a surplus through negotiation with their partner. The assurance that investments will yield more than default profiting scenarios alleviates the hesitation that leads to underinvestment.
Understanding and addressing the hold-up problem is critical in forming stable partnerships. It ensures that all parties are not only willing but also eager to contribute their fair share towards an investment, guided by the confidence of a fruitful partnership outcome.

Investment Decisions

Investment decisions in strategic game scenarios are pivotal, especially when partners evaluate the potential benefits and drawbacks of allocating resources into a venture. The business partnership exercise presents this decision-making process in a clear framework.
Players must decide whether the investment with an upfront cost of 3 outweighs the potential profits shared after forming a partnership. These decisions become crucial since they affect the formation of the partnership and the profitability of involved players. The decision rule, aided by conditions like "x > 9," directs players to contribute investments as long as the expected payoff surpasses alternative scenarios.
This calculation of potential returns is fundamental to sound investment decisions, fostering mutual benefits and cooperative strategies. It encourages players to look beyond immediate costs and focus on long-term advantages, ensuring that shared investments result in both personal and joint success.