The Black Friday Prisoner’s Dilemma
Since 2005 Black Friday has been the busiest shopping day in the United States [3], where holiday shopper attempt to get gifts early at bargain prices. It has been customary for years for retailers to open early, but recently opening times has been taken to new extremes. It used to be customary (around 2005) for retailers to open a bit earlier, around 6am, to help space out the shoppers, by having the very eager shoppers arrive first. Later in 2000’s stores began opening a bit earlier, around 4am/5am. Last year, for the first time, retailer pushed it to the limit, by starting the sales as soon as Friday started and opening at midnight. This included several major retailers such as Target, Kohls, Macy’s, Best Buy, and Bealls [3]. This years some retailers, including Walmart, Target, and Toys R Us [1], have completely tossed the “Friday,” and are opening as early as 8pm on Thursday, Thanksgiving Day. As the times become earlier and earlier, more and more people are inconvenienced and it results in increased costs retailers. This raises the question as to why. The answer can be explained using a bit of elementary game theory, which we will discover is a case of a prisoner’s dilemma.
To start, we shall establish a way to model the payoffs for all of the stores. As stores open earlier, the scarcity of willing employees to work increases, and so does the cost of hiring employees to work for the sale. Furthermore, typically night time wages are larger than daytime wages. As the employee’s salary increases, costs increases for retailers. Furthermore, if stores open earlier overall less people (fewer willing to sacrifice thanksgiving meal, and those some that would normally show up later may not show up due to feeling some of the deals may be gone). Thus revenue also decreases. However, while this may be true, it complicates the model, and does not affect the resulting analysis, so we will assume that the total number of customers is constant. Overall, as costs increase and revenue decreases, profit decreases.
For the sake of simplicity, we will assume that customers all find stores equally as attractive and do not have a retailer preference. Furthermore, we will divide the time into intervals of hours, and assume that each customer chooses 1 hour to shop in with equal probability. (Thus the 3rd hour has just as many customers as hour 5). Thus, at a given time t and time spent open n, if St stores are open, and there are Ct,n customers out, then a store will have Rt,n=Ct,n/St customers. (Ct,1>Ct,2 as in the first case stores opened fewer hours (and thus later)). Finally, we assume that the number of customers in a store is directly proportional to the sales or revenue. Thus, we can model revenue by being Rt,n, and loss or cost to be Lt.
If all store decide not open early, they will earn ∑Rt,n-Lt for t=0 to n hours. However, all the stores decide to open earlier, they will all have less profit as there are increased cost and total revenue remained the same: Cn+1,n+1/Sn+1-Ln+1+∑Rt,n+1-Lt for t=0 to n hours. However, if only one store decides to open an hour earlier, then they will get an advantage over all the other stores by getting all the customers out during the first hour, and splitting the remaining customers: Cn+1,n+1/1-Ln+1+∑Rt,n+1-Lt for t=0 to n hours. Thus we can model the Black Friday retailer game as
| All other stores→ Store X ↓ | NOT Early | Early |
| NOT Early | 100, 100 | 85, 105 |
| Early | 120, 92 | 95, 95 |
This game represents the intuition developed above, and we focus on relative earnings between choices, not the exact numbers. The first entry represents the earning for store X, the second entry represents the earning for all other stores. If a store is the only one to open early it profits the most. If all stores open early, they profit less than if they all did not. Say the year is 2009 and all stores opened at 5am. Then store X will note that if all other stores do not go earlier, then X is best off going at early, as they will have a huge increase in sales. Even, if all the other stores go early, X does not want to be left behind and have the least profit, so it is better to go early. Thus, it is always better for X to go early, and we can say going Early strictly dominates NOT Early. Similarly every store looking at the payoff matrix will conclude the same, thus it is always better to go early. However, since everyone went early, everyone got less profit, and we have our Prisoner’s Dilemma.
If everyone had been able to cooperate and decide to open later, it would have resulted in higher earnings for all companies (not to mention happier consumers who didn’t have to cut into their thanksgiving meal to go shopping). However, the only Nash equilibrium of the game is for everyone to go early (it is always beneficial (the best response) for someone going not early to switch to early, thus all other possibilities are not Nash equilibrium). Early for everyone is a Nash equilibrium, as no one person switching can make themselves better off, so everyone remains early (best response is early for everyone). Thus, in conclusion, we unfortunately have situation where retailers and consumers are all worse of this holiday season. Happy Holidays!
-esh
Sources
[2] http://www.charlotteobserver.com/2012/11/14/3663509/stores-open-for-black-friday-earlier.html
[3] http://en.wikipedia.org/wiki/Black_Friday_(shopping)