Question

Consider a two-player contractual setting in which the players produce as a team. In the underlying game, players 1 and 2 each select high (H) or low (L) effort. A player who selects H pays a cost of 3 ; selecting \(\mathrm{L}\) costs nothing. The players equally share the revenue that their efforts produce. If they both select \(\mathrm{H}\), then revenue is 10 . If they both select \(\mathrm{L}\), then revenue is 0 . If one of them selects \(\mathrm{H}\) and the other selects \(\mathrm{L}\), then revenue is \(x\). Thus, if both players choose \(\mathrm{H}\), then they each obtain a payoff of \(\frac{1}{2} \cdot 10-3=2\); if player 1 selects \(\mathrm{H}\) and player 2 selects \(\mathrm{L}\), then player 1 gets \(\frac{1}{2} \cdot x-3\) and player 2 gets \(\frac{1}{2} \cdot x\); and so on. Assume that \(x\) is between 0 and 10 . Suppose a contract specifies the following monetary transfers from player 2 to player \(1: \alpha\) if \((\mathrm{L}, \mathrm{H})\) is played, \(\beta\) if \((\mathrm{H}, \mathrm{L})\) is played, and \(\gamma\) if \((\mathrm{L}, \mathrm{L})\) is played. (a) Suppose that there is limited verifiability in the sense that the court can observe only the revenue \((10, x\), or 0\()\) of the team, rather than the players' individual effort levels. How does this constrain \(\alpha, \beta\), and \(\gamma\) ? (b) What must be true about \(x\) to guarantee that \((\mathrm{H}, \mathrm{H})\) can be achieved with limited verifiability?

Short answer

To achieve (H, H), set transfers to minimize deviation incentives, and ensure \(x > 7\).

Step-by-step solution

Analyze revenue and effort outcomes

First, examine the different combinations of effort and their corresponding revenues. If both players choose High (H), the revenue is 10 and each player's payoff is \(\frac{1}{2} \times 10 - 3 = 2\). If both choose Low (L), the revenue is 0 and payoff is 0 for each. If one chooses H and the other L, revenue is \(x\) and payoffs depend on the chosen contract terms.

Understand contract constraints due to verifiability

Since the court can only observe total revenue (either 10, \(x\), or 0), the transfers \( \alpha, \beta, \gamma \) can only depend on these values. Thus, \(\alpha\) applies when revenue is \(x\) and was generated by (L, H), \(\beta\) when revenue is \(x\) from (H, L), and \(\gamma\) when revenue is 0.

Set conditions to achieve (H, H) with contracts

To ensure (H, H) as an equilibrium, both players need a higher payoff from (H, H) than any (L, H) or (H, L) mix. Thus, the payoff for (H, H) must satisfy 2 > payoff from any deviation incentives, considering possible transfer payments. Also, since \(\alpha\) and \(\beta\) are unknown, setting appropriate such that it doesn't incentivize deviations over staying (H, H).

Solve for x in (H, H) equilibrium condition

For (H, H) to be Nash equilibrium, neither player should benefit more by unilaterally changing strategy. So, for player 1:\[2 \geq \frac{1}{2} x - 3 + \beta\]and for player 2:\[2 \geq \frac{1}{2} x - 3 + \alpha\]Simplifying both inequalities gives a condition on \(x\) and assuming transfers minimize deviations between (H, H) and unilateral (L, H) or (H, L).

Check constraints on x

Since \(x\) is between 0 and 10, ensure that the value satisfies these derived conditions, typically requiring \(x > 7\) to maintain a payoff threshold considering transfer incentivization for players.

Key concepts

Nash Equilibrium

In game theory, a Nash Equilibrium is a situation where no player can benefit by changing their effort level, assuming other players keep their strategy unchanged. In the context of our exercise, to achieve Nash Equilibrium, the players must coordinate in a way that neither prefers switching from their chosen effort level.
This means if both players pick High (H) effort, they should collectively secure the highest possible personal payoff.

• Choosing High (H) results in a revenue of 10 shared equally. Therefore, each gets a payoff of 2 after costs.
• If either player deviates from H to Low (L) while the other remains High, their payoff might increase momentarily. However, for equilibrium, any benefit from deviation should be mitigated.

This can be achieved by setting up conditions that discourage deviations, ensuring the most beneficial outcomes are preserved when both stay with (H, H).

Contractual Setting

The contractual setting defines the terms according to which both players are compensated based on team revenue. In this scenario, players have pre-determined contracts that stipulate monetary transfers based on different outcomes, driven by revenue observations rather than individual efforts.
The constraints of a contract in this setting involve transfers denoted by \( \alpha \), \( \beta \), and \( \gamma \):

• \( \alpha \) is applicable when the revenue \(x\) arises from one player exerting Low (L) effort and the other High (H).
• \( \beta \) is assigned when the revenue \(x\) is due to the inverse effort configuration, (H, L).
• \( \gamma \) is relevant when both players select Low (L), resulting in zero revenue.

With only team revenue verifiable, these parameters must be strategically set to align incentives and maintain commitment to the equilibrium without deceptive advantages.

Effort Levels

Effort levels represent the actions each player can take in the game. Players have the option to choose between a high effort (H) or low effort (L). The level of effort affects the team's revenue and individual payoffs.
High (H) effort incurs a cost for the player, but potentially yields higher overall revenue. This creates a strategic decision-making challenge because

• Choosing High (H) leads to a cost of 3 but, assuming reciprocity, generates a payoff of 2 each for the players.
• Selecting Low (L) has no costs but can result in lower payoffs overall, particularly if done unilaterally.

Players need to gauge whether the cost of effort is justified by the resulting benefit, both in terms of personal payoff and the additional benefits from adhering to established equilibrium principles.

Monetary Transfers

Monetary transfers are crucial in this setup for incentivizing players to maintain desired effort levels. They act as financial inducements, ensuring players behave in ways that support equilibrium while acknowledging the limits of observable effort.

• \( \alpha \), \( \beta \), and \( \gamma \) function to penalize or reward actions based on revenue results, which are the only observable metric.
• Transfers must be set carefully to deter players from breaking equilibrium; too high a transfer on deviations may lead to unnecessary preferred shifts, undermining the contract's integrity.
• Correctly aligned transfers should encourage players to select (H, H), ensuring optimal revenue and thus, maximal shared payoff.

In this context, monetary transfers function not only as rewards but also as mechanisms to regulate behavior, balancing between personal interest and shared profit.