Question

In: Economics

The year is 1870, the location is the town of Silverton in Colorado. There are two...

The year is 1870, the location is the town of Silverton in Colorado. There are two saloons in town: Red's Beard and Sadie's White Garter. After years of cut-throat competition the two owners, Red and Sadie, decide to cooperate in order to make more money. They have been making $400 per month each by competing fair and square. They decide if they each raise prices and limit the number of beers served, they will earn $1,000 per month each. However if one limits the number of beers and raises prices and the other does not, then they will earn only $200 per month while their competitor will earn $1,500 per month. Not knowing what to do, they turn to you, the prospecting game theorist, to help them figure out what to do.

Fill in the following table with the payoffs they can expect. Enter as follows: (Red's payoff, Sadie's payoff). Enter whole numbers - no commas.

Sadie's
Raise Prices Do Not Raise Prices
Red's Raise Prices (                            [ Select ]                       ["400", "1,000", "1,500", "200"]         ,                            [ Select ]                       ["1,500", "1,000", "200", "400"]         ) (                            [ Select ]                       ["400", "1,500", "1,000", "200"]         ,                            [ Select ]                       ["200", "1,500", "400", "1,000"]         )
Do Not Raise Prices (                            [ Select ]                       ["1,500", "200", "1,000", "400"]         ,                            [ Select ]                       ["1,000", "400", "200", "1,500"]         ) (                            [ Select ]                       ["400", "1,000", "1,500", "200"]         ,                            [ Select ]                       ["1,500", "400", "1,000", "200"]         )

Based on the payoffs, what is the likely outcome of the game? Explain

both of them raises prices

both of them do not raise prices

sadie does but red does not

red does but sadie does not raise prices

Solutions

Expert Solution

The problem seems to have an uncanny similarity with the classic 'Prisoners Dilemma' problem. Not surprising. As the situation stands, it seems that cooperation could give them both a steady income but, if one of them breaks the deal and takes advantage, there is a good amount of money to be earned. ( also friendship will be lost over beer, oddly) Then again both of them are in a similiar position to take advantage and if they both try to take advantage, they will lose a lot of  money, becoming competitors,. The payoff matrix makes the story a bit more clear:

red(first)/sadie(next) raise prices do not raise prices
raise prices 1000,1000 200,1500
do not raise prices 1500,200 400,400

It seems that if both Red and Sadie cooperate with each other and raise prices, they will be doing good earning 1000 each. It seems like a good deal. The only problem is that in that case, each of them will have an incentive to under-cut the prices and lure away the customers to gain more (from 1000 to 1500). Now if one cuts the price, the other will follow suit and then with successive undercuts, they will be back to being competitors earning a paltry 400 each. Not raising is the equililibrium strategy for both but they will only get sub-optimal results. The dilemma of this game is that both have an incentive to under-cut each other to gain more but sadly, since both take the same path to retaliation they end up being worse off then thwy would be if they could keep their promise.


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