Question

Suppose that Shtinki Corporation operates a chemical plant, which is located on the Hudson River. Downstream from the chemical plant is a group of fisheries. The Shtinki plant emits some byproducts that pollute the river, causing harm to the fisheries. The profit Shtinki obtains from operating the chemical plant is a positive number \(X\). The harm inflicted on the fisheries due to water pollution is measured to be \(Y\) in terms of lost profits. If the Shtinki plant is shut down, then Shtinki loses \(X\) while the fisheries gain \(Y\). Suppose that the fisheries collectively sue Shtinki Corporation. It is easily verified in court that Shtinki's plant pollutes the river. However, the values \(X\) and \(Y\) cannot be verified by the court, although they are commonly known to the litigants. Suppose that the court requires the fisheries' attorney (player 1) and the Shtinki attorney (player 2 ) to play the following litigation game. Player 1 is supposed to announce a number \(y\), which the court interprets as the fisheries' claim about their profit loss \(Y\). Player 2 is to announce a number \(x\), which the court interprets as a claim about \(X\). The announcements are made simultaneously and independently. Then the court uses Posner's nuisance rule to make its decision. \({ }^{9}\) According to the rule, if \(y>x\), then Shtinki must shut down its chemical plant. If \(x \geq y\), then the court allows Shtinki to operate the plant, but the court also requires Shtinki to pay the fisheries the amount \(y\). Note that the court cannot force the attorneys to tell the truth. Assume the attorneys want to maximize the profits of their clients. (a) Represent this game in the normal form by describing the strategy spaces and payoff functions. (b) For the case in which \(X>Y\), compute the Nash equilibria of the litigation game. (c) For the case in which \(X

Short answer

(a) Strategy spaces are real numbers for both players. (b) Equilibrium: Player 1 claims \(y = Y\), Player 2 claims \(x = Y + \epsilon\). (c) Equilibrium: Player 1 claims \(y > X\), Player 2 claims \(x \leq X\). (d) Rule is inefficient when \(X < Y\).

Step-by-step solution

Define Strategy Spaces

The strategy spaces for player 1 (fisheries' attorney) and player 2 (Shtinki's attorney) consist of all possible claims they can make about their respective profit losses and gains. Player 1's strategy space is all possible values of \( y \) they can claim as the harm to fisheries. Similarly, player 2's strategy space is all possible values of \( x \) they can claim as the profit from operating the plant.

Define Payoff Functions

For player 1, if \( y > x \), the plant shuts down and the payoff is \( Y \). If \( x \geq y \), the plant operates, and they receive \( y \) as compensation, making the payoff \( y \). For player 2, if \( y > x \), they receive a payoff of \( 0 \) as they must shut down. If \( x \geq y \), their payoff is \( X - y \) because they can operate and must pay \( y \) to the fisheries.

Analyze Case \( X > Y \) for Nash Equilibria

In this scenario, player 2 can always claim a number \( x \) slightly greater than \( y \) to ensure \( x \geq y \) because operating the plant provides a greater payoff. The equilibrium occurs when player 1 claims \( y = Y \) and player 2 claims \( x = Y + \epsilon \), where \( \epsilon \) is a small positive number. The payoffs become \( (Y, X - Y) \), maximizing interest for player 2.

Analyze Case \( X < Y \) for Nash Equilibria

Here, shutting down the plant would actually maximize player 2's loss minimization as paying \( y \) for operations would exceed their profit. The equilibrium is when player 1 claims any \( y > X \) and player 2 claims any \( x \leq X \). The plant shuts down, resulting in payoffs \( (Y, 0) \). Player 1 maximizes their payoff, while player 2 minimizes their loss by not exceeding the profit \( X \).

Analyze Efficiency of the Court's Rule

The court's rule is considered efficient if it results in the maximum combined payoffs for both parties. In the case \( X > Y \), the rule allows efficient operation of the plant. However, in the case \( X < Y \), the plant erroneously continues to operate if strategies don't meet Nash Equilibrium, making the rule inefficient in scenarios of misinformation.

Key concepts

Nash Equilibrium

In game theory, **Nash Equilibrium** is a fundamental concept that describes a situation where no player can benefit by changing their strategy while the other players keep theirs unchanged. Consider a game involving two attorneys, where each represents opposing interests: the fisheries and a chemical plant. Both players announce strategies simultaneously without knowing the opponent's choice. When both players choose strategies such that neither one benefits from unilaterally changing their announced values, the game reaches a Nash Equilibrium.
The Nash Equilibrium helps predict the outcome of strategic games. When the plant's profit exceeds the loss to fisheries ( X > Y ), the plant attorney might always choose a slightly higher value. This approach ensures the plant's continued operation and results in optimal payoffs for the plant. Conversely, if the plant's profit is less than the fisheries' loss ( X < Y ), shutting down the plant becomes the chosen strategy, making Nash Equilibrium essential to understanding and predicting players' actions.

Normal Form Game

A **Normal Form Game** is a representation used in game theory to simplify and analyze strategic interactions among players. In this format, each player has a defined strategy space, and their payoffs are depicted as a function of their chosen strategies and those of their opponents.
This game involves two players: the fisheries' attorney and the plant attorney. Each has a set of strategies, forming the strategy spaces (y) and (x) , representing potential claims about losses and gains. The game board would list these strategies with corresponding payoffs, crafted through their strategic choices. The optimal way to play is determined by predicting the opponent's move and adopting the best response. The normal form helps us map out the intricate dynamics of decisions in pursuit of maximizing individual client profits in litigation.

Payoff Functions

In any strategic game, **Payoff Functions** describe how outcomes return value to players based on their chosen strategies. Each attorney, representing either the fisheries or the plant, uses these functions to assess the gains or losses from the strategies selected.
Player 1 (the fisheries' attorney) has two scenarios:

• If claimed harm (y) is greater than profit claimed by the plant ( x ), the payoff is (Y) due to the plant's shutdown.
• If not, they gain compensation, quantified as (y) .

Player 2 (the plant attorney) similarly has two scenarios:

• The plant is operable without payout (zero payoff) when (y > x) .
• Otherwise, the payout is based on (X - y) , subtracting compensation to fisheries from the plant's profit.

These payoff functions aid in analyzing strategic decisions and their impact on outcomes based on value optimization.

Strategy Spaces

**Strategy Spaces** define the possible actions available to each player in the game. For attorney players in our scenario, these spaces encompass all conceivable claims about profit losses and gains. This range of strategies is depicted as potential numerical values that attorneys can select.
Player 1 can choose any value of (y) reflecting losses from pollution, while Player 2 chooses (x) to represent the plant's profitability claims. Each strategy endeavor is crafted to optimize outcomes, considering both competitive and compensatory objectives in a legal contest. Understanding these spaces allows players to explore various outcomes and form strategic plans that best uphold their clients' interests while navigating complex litigation scenarios.