Question

Consider a game between a police officer (player 3 ) and two drivers (players 1 and 2). Player 1 lives and drives in the Clairemont neighborhood of San Diego, whereas player 2 lives and drives in the Downtown area. On a given day, players 1 and 2 each have to decide whether or not to use their cell phones while driving. They are not friends, so they will not be calling each other. Thus, whether player 1 uses a cell phone is independent of whether player 2 uses a cell phone. Player 3 (the police officer) selects whether to patrol in Clairemont or Downtown. All of these choices are made simultaneously and independently. Note that the strategy spaces are \(S_{1}=\\{\mathrm{U}, \mathrm{N}\\}, S_{2}=\\{\mathrm{U}, \mathrm{N}\\}\), and \(S_{3}=\\{\mathrm{C}, \mathrm{D}\\}\), where "U" stands for "use cell phone," " \(\mathrm{N}\) " means "not use cell phone," "C"' stands for "Clairemont," and "D" means "Downtown." Suppose that using a cell phone while driving is illegal. Furthermore, if a driver uses a cell phone and player 3 patrols in his or her area (Clairemont for player 1, Downtown for player 2), then this driver is caught and punished. A driver will not be caught if player 3 patrols in the other neighborhood. A driver who does not use a cell phone gets a payoff of zero. A driver who uses a cell phone and is not caught obtains a payoff of 2 . Finally, a driver who uses a cell phone and is caught gets a payoff of \(-y\), where \(y>0\). Player 3 gets a payoff of 1 if she catches a driver using a cell phone, and she gets zero otherwise. (a) Does this game have a pure-strategy Nash equilibrium? If so, describe it. If not, explain why. (b) Suppose that \(y=1\). Calculate and describe a mixed-strategy equilibrium of this game. Explain whether people obey the law. (c) Suppose that \(y=3\). Calculate and describe a mixed-strategy equilibrium of this game. Explain whether people obey the law.

Short answer

(a) No pure-strategy Nash equilibrium. (b) Mixed-equilibrium: drivers often disobey law when \(y=1\). (c) Mixed-equilibrium: better law compliance when \(y=3\).

Step-by-step solution

Analyze Pure-Strategy Nash Equilibrium (Part a)

To find a pure-strategy Nash equilibrium, we examine all possible strategies for the players. Consider the strategies: (1) both drivers not using the cell phone, police patrols either location; (2) both drivers use the cell phone and get caught; (3) one driver uses the cell phone and gets caught while another avoids. However, if both use the cell phone and get caught, they both receive negative payoffs, leading them to prefer strategies that bring non-negative payoffs. A pure strategy equilibrium doesn't exist since players will always find advantageous deviations.

Calculate Mixed-Strategy Equilibrium for y = 1 (Part b)

When the fine is 1, driver payoff when caught is -1. For player 1 to mix his strategies, his expected payoff from using a cell phone should equal not using it. Assuming the police patrol Clairemont with probability \(p\), and Downtown with probability \(1-p\), the payoff equations are: \[2(1-p) - 1(p) = 0\]For the police not to prefer Clairemont, the expected payoffs for both drivers must be equal. Similarly, the probability \(q\) of each driver using the phone could be determined by police outcomes: \(q = 2/3\). This makes both the police and drivers indifferent.

Analyze Implications for Obeying Law at y = 1 (Part b)

In this mixed-strategy equilibrium for \(y=1\), the probability of using a cell phone is significant (i.e., \(q = 2/3\)), which means drivers frequently disobey the law due to relatively low penalties.

Calculate Mixed-Strategy Equilibrium for y = 3 (Part c)

Now consider \(y=3\). When caught, drivers incur a payoff of -3. Adjusting the equation for the driver's decision using police probability \(p\): \[2(1-p) - 3p = 0\]Solving gives us \(p = 2/5\). Similarly, resolving for police choosing probabilities given driver choices (using mixed strategy balancing): \(q = 3/5\). Here, both drivers and police have balanced incentives.

Analyze Implications for Obeying Law at y = 3 (Part c)

With \(y = 3\) and \(q = 3/5\), the likelihood of using a phone decreases compared to \(y=1\), suggesting that higher penalties lead to increased law compliance.

Key concepts

Nash Equilibrium

In the game involving two drivers and a police officer, we explore the concept of Nash Equilibrium to determine if there exists a set of strategies where no player can benefit by changing their strategy, given the other players' strategies remain the same. A pure-strategy Nash Equilibrium occurs when each player chooses a single, non-random strategy. However, in this scenario, pure-strategy Nash Equilibrium does not exist.

Why? If both drivers decide to use their cell phones and end up getting caught by the police officer, they will face negative payoffs due to the fines. This negative outcome motivates them to change strategies to avoid such penalties. If instead both choose not to use cell phones, the police patrol choice is irrelevant to their payoff. Hence, there isn't a single strategy combination that benefits all in a manner that no player wants to deviate from. As a result, players continuously seek better outcomes, and a pure-strategy Nash Equilibrium fails to materialize.

Mixed-Strategy Equilibrium

When pure strategies do not stabilize as equilibrium points, we turn to mixed strategies. A mixed-strategy equilibrium involves players randomizing over their available strategies to make other players indifferent to their choices. Each player adjusts the probability of their various strategies to keep others from gaining a strategic advantage by changing their tactics.

In our game scenario, we calculated mixed-strategy equilibria for different penalties. For a fine (\(y = 1\)), players choose whether to use their phones or not based on the probability the police will patrol their area. The calculations show drivers use their phones with probability \(q = 2/3\), and police patrol Clairemont with probability \(p = 1/3\). When the fine increases to \(y = 3\), \(q = 3/5\) and \(p = 2/5\). Mixed-strategy equilibria ensure that each player's decision is optimized based on others' likely choices, maintaining a balance where players have no clear incentive to change their strategies.

• Players are indifferent about using or not using a strategy when mixed-strategy equilibrium is achieved
• Probabilities are adjusted based on the penalty's severity

Law Compliance

The main question is whether individuals follow the law based on the penalties for violations. By examining the mixed-strategy equilibria, we find that compliance levels correlate with the severity of the penalties.

For example, with a fine of \(y = 1\), drivers disobey the law more frequently, given the probability \(q = 2/3\) of using their phones. This low penalty demonstrates insufficient deterrence, as the benefits outweigh the potential costs. However, increasing the penalty to \(y = 3\) changes compliance behavior significantly. The probability of using a phone drops to \(q = 3/5\), showcasing a stronger adherence to legal regulations due to heightened penalties.

• Higher fines reduce law violation probabilities
• Efficient law enforcement complements penalty effects

The repercussions of penalties in game theory suggest that player behavior aligns more with law compliance when facing substantial consequences for illegal acts.