# Big Players in a Small Market – Pure and Mixed-Strategy Nash Equilibria in Stylized Market Participation Decisions

## Summary

In the context of an oligopolistic entry game, students will translate a word problem into game theoretic notation and will then examine how changes to the payoff structure changes the Nash equilibria in a game. They will also solve for and discuss the Nash equilibrium in mixed strategies.

## Context for Use

● The activity is appropriate for both Principles and Intermediate Microeconomics. If mixed strategies are not covered in your Principles class, the latter portion of the problem can be removed, cutting the activity down by about 10 minutes.

● Students should have studied Nash equilibria in both pure and mixed strategies.

● This activity should be suitable for discussion in any class size.

● The activity will likely take around 30 minutes. Figuring out how the distribution of revenue in the market affect the Nash in pure strategies should take about 60% of that time.

## Overview

In this activity students will examine how the Nash equilibria change based on the payoff structure. Furthermore, students will learn how to think about situations where the Nash equilibrium in pure strategies is not unique and a single prediction as to how the game will play out is not possible. It is also meant to provide students with practice in translating a word problem which is not explicitly phrased as a game theory question (no outright payoff matrix) into a more formal structure that can be solved. This is all done in the context of an oligopolistic simultaneous entry game.

The second half of the exercise is meant to provide a practice/discussion of Nash equilibria in mixed strategies. It also highlights how the nature of the game changes with a small change to the payoff matrix (in this case it becomes a Hawk-Dove game). The main benefit of the exercise is having students actually talk through a mixed strategy equilibrium in a group.

## Expected Student Learning Outcomes

In this exercise students will fortify their understanding of the concept of Nash equilibria in pure and mixed strategies. They will also practice their skill at translating stylized word problems into economic models they are familiar with.

## Information Given to Students

PART 1:

Two large fast-food chains, Chipotle Mexican Grill and Qdoba Mexican Eats, are considering opening their restaurants in a small town. Both fast-food chains have similar technologies. Building, equipping, and staffing a new restaurant will cost either company one million dollars (\$1M). If both chains open their restaurants simultaneously, the potential total fast-food revenue of two and a half million dollars (\$2.5M) (for that year) would have to be split between them, with proportions depending on the preferences of the consumers in town. If a chain does not open a restaurant, it keeps the one million for other business purposes. Both chains care greatly about the short-term performance of the restaurants upon opening (this can be measured as the money in the bank account).

From the list below, pick the outcomes which are feasible. What conditions would lead to each of these outcomes?

A. Chipotle will build a restaurant while Qdoba will not.
B. Qdoba will build a restaurant while Chipotle will not.
C. Both Chipotle and Qdoba will build their restaurants.
D. Neither of the chains will choose to enter the market.

PART 2:
Now, suppose that the location researchers for both companies report that if both companies choose to enter the market simultaneously, the costs of building a restaurant would go up by a quarter of a million (\$0.5M) due to the need to hire extra workers from out of town. Furthermore, both marketing divisions report that it is most likely that if both companies are present in town, customers are equally likely to eat at each of the restaurants. Given this, can you predict the outcome of the interaction between Chipotle and Qdoba with certainty? What would be the optimal approach that the companies might take in this case?

Student Handout for "Big Players in a Small Market – Pure and Mixed-Strategy Nash Equilibria in Stylized Market Participation Decisions" (Microsoft Word 2007 (.docx) 17kB Aug26 19)
Instructor Solutions for "Big Players in a Small Market – Pure and Mixed-Strategy Nash Equilibria in Stylized Market Participation Decisions" (Acrobat (PDF) 52kB Aug26 19)

## Teaching Notes and Tips

● This activity was created in 2019. Chipotle and Qdoba can be replaced by any other two big/popular fast food franchises to keep the activity more relevant to students.

● What prefatory remarks should set up the application exercise? – If you want to make the exercise a bit easier, it may be a good idea to tell students that they might benefit from putting the profits for the two companies in an appropriate payoff matrix (do not tell them more than this) and to pay attention to what the costs are. However, this may be too much guidance on the offset and take away from the objective of translating a word problem into game theoretic structure.

● Does facilitating the teamwork require any special action? – No, beyond the usual circulation and making sure that the groups are actively discussing the problem at hand. A good approach would be to have students complete Part 1 and have a discussion about the equilibria and what distributions of revenues induced them. The first step of the discussion, after the students revealed their answers, should be reaching a consensus regarding what the payoff matrix for the game looks like. It would be a good idea to write down the (Enter, Enter) cell payoffs as functions of the preference parameter. In Part 1 there will only be equlibria in pure strategies, with options A, B, and C all being feasible depending on which franchise gets a bigger portion of the revenues. D is never an equilibrium. This should take about 20 minutes. Only then proceed to Part 2.

● What kinds of follow-up questions are recommended for facilitating the debriefing conversation among team reporters? In particular, how might the instructor get teams to evaluate which answer is the best, provide the analytical support for the team answers, identify what information would enable an economist to decide between alternative answers? - Turn the activity on its head: have the students think about the changes in the market (in this case, preferences) that can greatly impact the strategic behavior of companies. In this case slight shifts in preferences for one chain over the other result in going from one unique Nash equilibrium to its opposite (in terms of which chain enters the market). Also, note or emphasize the fact (depending on how much mention it received from your students) that both firms not entering is NEVER a Nash equilibrium in this game, which is quite interesting from the "real-world" standpoint.

● What points should be emphasized in the instructor's summary remarks to conclude the exercise? - Having discussed the above, it might be good to talk about the limitations of predictions given by stylized models, as well as talk about the many refinements of Nash equilibrium used in the field of Game Theory.

## Assessment

Having students solve any variation of 2-by-2 game with a mixed strategy Nash as a solution (e.g., a zero-sum game) can be used to check the mechanical understanding of concept. Having the students explain what players do in mixed-strategy Nash and what the outcomes would be in their own words could also be beneficial.

N/A