Question

Imagine that a zealous prosecutor (P) has accused a defendant (D) of committing a crime. Suppose that the trial involves evidence production by both parties and that by producing evidence, a litigant increases the probability of winning the trial. Specifically, suppose that the probability that the defendant wins is given by \(e_{\mathrm{D}} /\left(e_{\mathrm{D}}+e_{\mathrm{P}}\right)\), where \(e_{\mathrm{D}}\) is the expenditure on evidence production by the defendant and \(e_{\mathrm{P}}\) is the expenditure on evidence production by the prosecutor. Assume that \(e_{\mathrm{D}}\) and \(e_{\mathrm{P}}\) are greater than or equal to 0 . The defendant must pay 8 if he is found guilty, whereas he pays 0 if he is found innocent. The prosecutor receives 8 if she wins and 0 if she loses the case. (a) Represent this game in normal form. (b) Write the first-order condition and derive the best-response function for each player. (c) Find the Nash equilibrium of this game. What is the probability that the defendant wins in equilibrium. (d) Is this outcome efficient? Why?

Short answer

(a) Strategies are expenditures; (b) Optimal \(e_D = e_P\); (c) Nash Equilibrium: \(e_D = e_P\), \(p = \frac{1}{2}\); (d) Efficient via balance.

Step-by-step solution

Represent the Game in Normal Form

In normal form, we express the game in a matrix, showing payoffs for different strategies of Defendant (D's evidence expenditure) and Prosecutor (P's evidence expenditure). Both select expenditure levels, impacting respective payoffs determined as follows: For D, payoff is \(-8 \cdot \frac{e_D}{e_D + e_P}\), and for P, payoff is \(8 \cdot \frac{e_P}{e_D + e_P}\). The game is continuous with each player choosing \(e_D\) and \(e_P\), with strategies being non-negative numbers.

Derive First-Order Condition for Defendant

The defendant's goal is to minimize expected costs: \(-8 \cdot \frac{e_D}{e_D + e_P}\). The first-order condition is obtained by differentiating this with respect to \(e_D\), yielding \(-8 \cdot \frac{e_P}{(e_D + e_P)^2} = 0\). However, as \(e_D\) grows, \(-8\) becomes less impactful. The optimization happens under constraints given \(e_P\).

Derive Best-Response Function for Defendant

Given P's evidence expenditure \(e_P\), D's best-response function is derived by solving the first-order condition: maximizing \(-8 \cdot \frac{e_D}{e_D + e_P}\) constrained by \(e_P\). This doesn't yield a straightforward solution as \(e_D\) affects equally through denominator; thus D's optimal strategy balances \(e_D = e_P\) to minimize overall loss.

Derive First-Order Condition for Prosecutor

The prosecutor seeks to maximize payoff \(8 \cdot \frac{e_P}{e_D + e_P}\). Differentiating with respect to \(e_P\), the first-order condition is \(8 \cdot \frac{e_D}{(e_D + e_P)^2} = 0\). Similar structure as defendant's, looking to maximize given output of strategy deals across \(e_D\).

Derive Best-Response Function for Prosecutor

In similar fashion, solve for \(e_P\) following differentiation imprisonment: \(8 \cdot \frac{e_D}{(e_D + e_P)^2} = 0\) hints prosecutor's consistent balance against \(e_D\) leading to equality with \(e_D\) for reaching top possibility.

Find Nash Equilibrium

At Nash equilibrium, neither party can unilaterally deviate to improve their payoff given the opponent's strategy. Equality derived in best responses indicates at \(e_D = e_P\) holds; equal expenditure equaling game balance.

Calculate Probability Defendant Wins in Equilibrium

In equilibrium, since \(e_D = e_P\), the probability that the Defendant wins is \(\frac{e_D}{e_D + e_P} = \frac{e_D}{2e_D} = \frac{1}{2}\).

Evaluate Efficiency of Outcome

Efficiency in strategies is aligned. Each player exacts leaps equitably; neither of deductions nor focus achieves outgrowth sewer immunity on trade strategy equals non-loss. Equilibrium boringly remains rooted at pace, handling immense time shares.

Key concepts

Nash Equilibrium

The core idea of Nash Equilibrium in game theory is a situation where, given the strategies of the other players, no player can benefit by unilaterally changing their own strategy. For our exercise involving the prosecutor and the defendant, Nash Equilibrium occurs when both parties choose their "evidence expenditure" levels optimally, such that any change in one party's expenditure, while holding the other constant, ends up diminishing their payoff rather than increasing it.
In this specific trial situation, we see that at equilibrium both the defendant and the prosecutor choose the same expenditure level, meaning their expenditures are equal. This balance of evidence production ensures that neither can improve their situation without the other also changing theirs.
Reaching this equilibrium involves understanding that strategies and payoffs are interdependent - each player's optimal move depends directly on the other player's decision. Therefore, Nash Equilibrium reflects a state of mutual best-response between interacting agents in the game.

Normal Form Representation

Normal form representation or matrix form is a way of displaying a game that captures the strategies and payoffs available to players in a matrix. This format helps us visualize the strategic interactions between players. In our case, the normal form representation does not use a simple matrix because the game is continuous rather than discrete. Instead, we consider expenditure levels (strategies) of both players, which affect their payoffs.
The prosecutor's payoff is derived as a function of her evidence expenditure relative to the total expenditure, while the defendant's payoff conversely diminishes as he spends more to bolster his case. Placing these payoffs in a matrix would involve numerous, continuous choices of expenditure, rather than a simple table of discrete payoffs.
The essence of normal form here lies in presenting the strategic choices and consequent payoffs in terms of their dependency on each other's expenditure decisions.

Best-Response Functions

The best-response function identifies the optimal strategy for a player, given the actions chosen by the other player. It serves as a guide to how a player should act to maximize their payoff, considering the opponent's strategy.
For the defendant, the best-response function is found by setting the expenditure level that minimizes his expected loss, based on the prosecutor's expenditure. Mathematically, this involves taking the derivative of his payoff function and solving for the point where changing expenditure doesn't further minimize loss.
On the other side, the prosecutor's best-response function is calculated similarly but focuses on maximizing her gains through evidence expenditure. The solution indicates a balance where each party ends up matching their evidence budget.
Both players aim to equate each other's evidence expenditures, illustrating an interdependent strategy where each player's best response aligns with the other's move.

Efficiency in Economics

Efficiency in economics, especially in a legal setting as represented in this exercise, involves maximizing desired outcomes with given resources. Here, we're interested in achieving an equilibrium where neither player can increase their payoff without expense to the other. This implies no resources are being wasted.
Analytically, the game ends in a state where evidence production by both parties reflects an efficient allocation: equilibrium is reached where any increase in one party’s expenditure would lead to unnecessary expense without offering any additional gain in trial outcome.
From a broader perspective, efficiency here illustrates that the game based evidence expenditure levels are rationally aligned, albeit one might argue it's inefficient in societal terms, as significant resources are spent purely to reach a stalemate for litigants. Nevertheless, given the constraints of the situation, this mutual best response ensures each player's resources are utilized in the most balanced manner possible.