Question

Consider a contractual setting with two players, where the underlying strategic interaction is given by the following "partnership game": This matrix describes the technology of the relationship. Suppose that before noncooperative interaction, the players sign an externally enforced contract specifying a transfer $t$ from player 2 to player 1 in the event that either $(\mathrm{H},\mathrm{L})$ or $(\mathrm{L},\mathrm{H})$ is played. The interpretation here is that the court can observe the total output of the relationship (which is either 8,12 , or 16) and is willing to enforce a transfer if output is 12 . However, the court enforces the transfer only if one or both players request enforcement. Each player who requests enforcement must pay a legal fee of $c$. The court only allows transfers less than or equal to 10 (from one player to the other) because this is the greatest payoff a single player can earn in the partnership. Consider a "grand game" in which the partnership game is one component. Play in the grand game runs as follows. First, the players interact by playing the illustrated partnership game. Then, if either (H, H) or (L, L) is the outcome, the grand game ends. However, if (H, L) or (L, H) is the outcome, then the players interact further by playing the following enforcement game: The players' payoffs in the enforcement game are added to their payoffs from the partnership game. (a) Suppose you are a policy maker who determines the fee $c$ and the contract parameter $t$. Under what conditions on $c$ and $t$ will (H, H) be played in a subgame perfect equilibrium of this game? That is, for what values of $c$ is there a number $t$ that induces cooperation? Remember that $t$ is constrained to be between $-10$ and 10 . (b) Explain your reasoning for your answer to part (a). In particular, what happens when $c=0$ ? (c) Suppose that between the partnership game and the enforcement game the players make a joint decision about whether to continue with the enforcement game or "settle out of court"; they can also make a monetary transfer at this time. When the players settle out of court, they avoid the court fee. The default decision is to continue with the enforcement game. Do you think $(\mathrm{H},\mathrm{H})$ can be supported in the negotiation equilibrium of this game? Explain your reasoning without explicitly solving this game.

Short answer

To ensure (H, H) in equilibrium, $c$ should be minimized and $t$ should align payoffs to favor cooperation. Cooperation is more likely with $c=0$ or out-of-court settlements, supporting (H, H) with effective negotiation.

Step-by-step solution

Understanding the Setup and Constraints

First, comprehend the partnership game's payoff matrix and the external enforcement of a contract involving a transfer, $t$, which is applicable when the outcome is $(H,L)$ or $(L,H)$. Note the restriction that transfer amounts $t$ range between -10 and 10, and both players must pay a fee $c$ to request enforcement in case of these outcomes. The objective is to determine values of $c$ and $t$ that support cooperation, $(H,H)$, as a subgame perfect equilibrium.

Analyzing Payoff Conditions for Subgame Perfect Equilibrium

For $(H,H)$ to be a subgame perfect equilibrium, the payoff from playing $H$ must be greater than or equal to playing $L$. This implies their payoff from $(H,H)$ and avoiding additional enforcement fees and outcomes should be greater than any potential gain or loss from $(H,L)$ or $(L,H)$ outcomes or from not requesting enforcement. This involves ensuring the combined payoffs from the partnership and enforcement games are maximized and net beneficial.

Evaluating Payoff for (H, H) without Enforcement Cost

When $c=0$, players avoid any cost if enforcement is requested. In this case, the payoff from playing $(H,H)$ should still exceed potential enforced payoffs minus $t$. This means either transfer should not incentivize non-cooperative outcomes through enforcement, so $t$ considered should balance payoff scenarios.

Assessment of Cooperation Support with Out-of-Court Settlements

If players can settle out of court, avoiding $c$, transfer decisions $t$ now include voluntary agreements. Cooperation is supported when all players receive higher payoffs, avoiding enforcement through negotiation, thus fostering $(H,H)$ as cooperative agreement without needing the court. The instability risk reduces as players favor optimal payoff scenarios from negotiation.

Key concepts

Subgame Perfect Equilibrium

In game theory, a Subgame Perfect Equilibrium (SPE) is a refined concept of Nash Equilibrium. It refers to a strategy profile that represents a Nash Equilibrium in every subgame of the original game. This means that the strategy must lead to optimal outcomes in every possible scenario following any play history.
In the context of the partnership game, SPE ensures cooperation as a stable outcome when both players choose strategies that maximize their payoffs in every possible sub-scenario. For the players to consistently choose $H,H$ as a strategy, the resulting payoffs must outweigh those from any other tactical variations such as $H,L$ or $L,H$ where eventual costly contract enforcements may come into play. This requires analyzing the cost $c$ and transfer $t$ settings that secure SPE in favor of cooperation, avoiding any inducement to diverge into non-cooperative play at any stage of the game.

Partnership Game

The Partnership Game is a strategic interaction involving two players with shared incentives and possible outcomes varying based on their actions. The game outlined involves players choosing strategies $H$ or $L$, with their combined actions producing varied outputs like 8, 12, or 16.
This scenario shares common traits with classical economic partnerships, where individual actions impact joint outcomes. In our exercise, successful plays like $H,H$ lead to higher joint outputs, theoretically encouraging cooperation. However, non-cooperative plays $H,L$ or $L,H$ could trigger enforcement mechanisms, affecting overall gains due to associated costs. Analyzing incentive structures, such as the payment of the legal fee $c$, becomes crucial as it influences decisions over whether to act collaboratively or pursue individualistic approaches.

Contract Enforcement

Contract Enforcement in this context involves implementing legal commitments pre-agreed by players, triggered in certain outcomes. Here, contract enforcement occurs if players end up with $H,L$ or $L,H$ results, leading to a transfer $t$ and potential enforcement cost $c$.
The court plays a vital role by ensuring only feasible monetary transfers under output scenarios. However, enforcement is contingent upon player-initiated requests, along with corresponding legal fees, making this a strategic decision point. If $c$ is set as zero, for example, enforcement costs disappear, altering payoff calculations and potentially skewing strategies towards immediate enforcement requests. Understanding the impact of enforcement in deterring or promoting non-cooperative behavior helps gauge adjustments like deciding optimal $t$ values beneficial in maintaining coopertive equilibrium or negotiation-driven resolutions.

Cooperative Strategies

Cooperative Strategies encourage players to work together, maximizing joint benefits rather than seeking individual gains at each other's expense. In the partnership game, achieving $H,H$ through cooperative strategy ensures higher overall payoffs than mixed or solitary approaches.
Cooperative strategies are fostered by structures that make deviation costly, such as potential court fees $c$ or strategically non-beneficial transfer terms $t$. Additionally, incorporating mechanisms like out-of-court settlements can bolster cooperation, allowing players to prevent further costs and preserve higher joint profits.
By prioritizing collaborative over adversarial tactics, participants are empowered to maintain mutually beneficial outcomes, thus reinforcing cooperation as the preferred course of action. Recognizing settings where cooperative dynamics are sustainable is key, allowing for strategies that can amicably resolve potential disputes or inefficiencies without necessitating third-party interventions.