Question

Heather and David (players 1 and 2 ) are partners in a handmade postcard business. They each put costly effort into the business, which then determines their profits. However, unless they each exert at least 1 unit of effort, there are no revenues at all. In particular, each player \(i\) chooses an effort level \(e_{i} \geq 0\). Player \(i\) 's payoff is \(u_{i}\left(e_{i}, e_{j}\right)= \begin{cases}-e_{i} & \text { if } e_{j}<1 \\\ e_{i}\left(e_{j}-1\right)^{2}+e_{i}-\frac{1}{2} e_{i}^{2} & \text { if } e_{j} \geq 1\end{cases}\) where \(j\) denotes the other player. (a) Prove that \((0,0)\) is a Nash equilibrium. (b) Graph the players' best responses as a function of each other's strategies. (c) Find all of the other Nash equilibria.

Short answer

(0,0) and (1,1) are Nash equilibria.

Step-by-step solution

Understand the Payoff Function

The payoff function for player \(i\) depends on both players' efforts, \(e_i\) and \(e_j\). If player \(j\) exerts less than 1 unit of effort, player \(i\)'s payoff is \(-e_i\). If player \(j\) exerts at least 1 unit, the payoff is \(e_i(e_j-1)^2 + e_i - \frac{1}{2}e_i^2\). This indicates that without minimum effort from both, the business earns nothing.

Prove (0,0) is a Nash Equilibrium

To prove \((0,0)\) is a Nash equilibrium, show that neither player can increase their payoff by unilaterally changing their effort from 0. For \(e_j < 1\), the payoff is \(-e_i\), which is minimized at \(e_i = 0\). Since \(0\) is the lowest effort yielding the highest possible payoff of zero, \((0,0)\) is a Nash equilibrium.

Identify Best Responses Conditional on Effort

When \(e_j \geq 1\), player \(i\)'s payoff is given by \(e_i(e_j-1)^2 + e_i - \frac{1}{2}e_i^2\). The best response for player \(i\) maximizes this expression. Set the derivative with respect to \(e_i\) to zero to find critical points: \((e_j-1)^2 + 1 - e_i = 0\), which implies \(e_i = (e_j-1)^2 + 1\) if \(e_j \geq 1\).

Graph Best Responses

Graph the function \(e_i = (e_j-1)^2 + 1\) for both players to find their best responses when \(e_j \geq 1\). Each graph indicates the player's optimal effort given the effort level of the other player, with intercepts at \(e_j=1\) resulting in \(e_i=1\).

Identify Other Nash Equilibria

A Nash equilibrium occurs where the best response functions intersect. For \(e_j \geq 1\), set \(e_1=(e_2-1)^2 + 1\) and \(e_2=(e_1-1)^2 + 1\). Solving \(e_1=e_2=1\) gives another solution. Therefore, the Nash equilibria are \((0,0)\) and \((1,1)\).

Key concepts

Game Theory

Game theory is a fascinating field of study that examines the behaviors of decision-makers, often referred to as "players," within strategic situations. In games, these players make choices simultaneously or in sequence, and these choices impact the outcomes for all involved. The goal in game theory is to predict and analyze these interactions efficiently.

• It involves studying how players choose strategies that maximize their payoffs while considering the actions of others.
• It provides a framework for understanding competitive scenarios, where each player's payoff depends not only on their own decisions but also on the decisions of others.
• Game theory has applications in economics, political science, biology, and more.

For Heather and David, game theory helps us explore how their efforts in the postcard business affect their individual and joint success. By using this theory, we can determine the strategic choices that they are likely to make, knowing they have to balance effort against the gains they receive.

Best Response

The concept of a "best response" is about choosing the optimal strategy given what other players are doing. In any strategic interaction, only certain actions will maximize a player's payoff based on the actions of others.

• A strategy is a best response if it results in the highest possible payoff, assuming the strategies of the other players are fixed.
• In the case of Heather and David, a best response would be the level of effort that maximizes each player's payoff, given the effort of their partner.
• Calculating the best response involves looking at the payoff function and using calculus to determine strategies that optimize it.

For Heather and David, identifying best responses helps them choose effort levels that either maximize individual payoffs or stabilize mutual gains.

Payoff Function

A payoff function is a crucial part of any game in game theory because it represents the rewards a player receives from choosing certain strategies. It shows how a player's utility or satisfaction depends on their own and others' actions.

• The payoff function often takes into account costs, benefits, and interactions with other players.
• For Heather and David, their payoff depends on both their effort levels: if either exerts less than one unit of effort, payoffs are negative; otherwise, it's a complex function involving their efforts.
• Understanding this function helps each player evaluate the consequences of different strategic choices.

By dissecting the payoff function, Heather and David can determine when their efforts will be rewarded and when they might not be, guiding them to more profitable decisions.

Strategic Interaction

Strategic interaction is the heart of game theory, where a player's strategy involves considering the reactions and actions of other players. This means that players must be forward-thinking and consider the ripple effects of their decisions.

• It involves anticipating how others will respond to one's actions.
• For Heather and David, strategic interaction requires understanding how different levels of effort will affect not just their own payoffs but also their partner's choices.
• Both players must strike a balance between competition and cooperation to maximize joint and individual outcomes.

Strategic interaction compels players to align their choices with expectations about their partner's actions, driving them towards Nash Equilibria, where neither player can benefit by changing their strategy unilaterally.