I’ve always wondered how the various coffee shops in Collegetown have determined their prices. Using what we’ve learned in class, can we model the Collegetown coffee “scene” with some auction and game-theoretic tools? And will this model accord with what we are observing in practice?

Setup:

There are four major coffee shops in Collegetown: Collegetown Bagels (CTB), Starbucks (SB), Wilson Farms (WF) and Eat Desserts First (EDF). Many more places sell coffee, but these are my usual destinations. My independent private values (IPV) for where I purchase coffee are a combination of the price of coffee and the distance I must go out of my way to buy coffee (really, this is just to say that more affects my preferences than just price):

IPV = 1 / ( price of coffee + distance )

My preferences can be thought of as a ranking of coffee shops, with my most preferred coffee shop(s) having the largest IPV. Currently, I prefer WF coffee because it is the least expensive and the least out of my way. I visit my preferred coffee shop(s) with a certain probability p > .7, because I do wonder what else is out there (1-p) of the time. However, since I’m trying to curb my caffeine addiction, I will only drink a single cup of coffee a day. If I have more than one preferred coffee shop, I will chose a destination from my preferred set with equal probability.

I am a very picky consumer and I like to think that all coffee shops exist to please me. More specifically, assuming all the coffee shops equally want to attract me to purchase their coffee, how would they set their prices?

Networked Coffee

We can think of the four coffee shops, CTB, WF, EDF and SB as Seller nodes in a bipartite graph with a single buyer, me. Currently, the preferred seller graph has a single edge between the buyer (me) and WF, as I prefer their coffee and no others. The three other sellers, assuming they cannot move locations, will try to increase my IPV for their coffee by lowering their prices. If their coffee’s IPV is equal to the maximum IPV, I will be equally happy (indifferent) between buying coffee at their establishment and some other preferred shops, and they will get more of my business than if I did not prefer them.

A perfect matching will be a set of market clearing prices. As in class and previous homework, we may create 3 fake “buyers”, for the purpose of matching each seller with a buyer.

A subtle point in this network is that the smaller the set of preferred shops, the greater the benefit for the preferred sellers. Since each preferred seller has an equal probability of being selected, the fewer the options, the greater patronage each seller will receive. Coffee shops will be motivated to lower their prices as far as possible because we will assume the patronage they will receive for being the single preferred seller is more important than the difference between their lowest prices.

Bean Bickering

On an individual level, each seller is motivated to lower their prices until they are in the preferred seller set. However, the coffee shops cannot arbitrarily lower their prices indefinitely. As in class, where houses were not sold for less than $0, we should cap the lowest price a particular coffee shop can charge. Since coffee proprietors are a crafty breed, they will only lower prices to their private “true” cost to produce the coffee, ensuring that they will never lose money on a sale. This is like an auction where sellers will sell their coffee at a price equivalent to the second lowest bid (which causes the second lowest seller to drop out), and therefore they will benefit from shading their true cost to produce coffee. Price wars can occur over many rounds and many different price postings. The winner’s dilemma applies here: if you “win” my business, you are valuing my business at a price below any of your competitors’ prices.

Secret Sourcing

What complicates matters further is that each seller now has a minimum price equal to their true sourcing cost – also an independent, private value. They are shading their prices, and they benefit when no one else knows their cost. My freedom in the coffee buying market closely resembles a network exchange model with perfect competition, where price wars between coffee shops may result in 0-profit sales at sourcing costs. With the slippery slope of network-exchange price wars, coffee shops must think carefully before lowering their prices. While lowering their price may add them to the preferred seller graph, it may also initiate a price war which they will lose, where the perfectly competitive price charged is lower than their true cost. If they lose the price war, their situation is worse than their intitially higher price at (1-p) frequency.

Since sourcing costs are secret IPV’s, the risk of initiating a network-exchange 0-profit price war is great. Much like the Prisoner’s Dilemma, all coffee shops are worse off if everyone lowers prices, while everyone is better off if no one lowers prices. If only one coffee shop lowers prices (becomes a preferred seller), that shop is maximally better off.

Stop, Drop, or Roll

It seems that in order to attract my business, coffee shops must evaluate the risks of lowering prices and losing money by engaging in a price war versus the risks of being in the not-preferred set. If coffee shops find the risks small enough, they will lower their prices incrementally, shading their true value to increase profit – going no lower than their true coffee-sourcing costs. They will either end up in a preferred seller graph, or hit their true value and remain in (1-p)-land. Remember, there is no guarantee that sourcing costs are unique. If coffee shops find the risks too high, they will hold prices relatively constant. Lowering prices seems to be “a roll of the dice” for coffee shops. In the market as I have described it, coffee shops are at the mercy of the consumer (me) unless they collude to fix prices.

Personally, this analysis seems to indicate that one should expect relatively stable coffee prices proportional to private costs and current trade volumes. Prices will not change much, for fear of fierce competition. Instead, we might expect proprietors to try and attract my business through non-price-lowering incentives, that would not trigger a price war. Indeed, this is supported by CTB’s new and unheard-of “Coffee Hour” discount!

Evidence of network markets at work!

Some questions for further thought:

1. Clearly not all price lowerings result in a price war, how can we fix our model?

2. How are initial prices for new shops picked?

3. Does current market power (in volume, price) effect network power?

4. What might be other non-price incentives?

An interesting, related problem is that of competitive facility location, if coffee shops can move their locations.

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