Pricing Strategy and Its Game Theory Applications
The author from the above article noted that Coke and Pepsi were never on sale at the same time. In fact, one academic paper has found that in a span of a 52week period, Coke and Pepsi ran nonoverlapping price promotions for 26 of those weeks. The odds of this happening by pure chance is 1/495,918,532,948,104, or in other words, extremely slim. The question now is why would those two competitors coordinate their pricing strategies when they should be fighting over market shares?
To answer this question, let us first look at the objective of a price promotion. The goal of a price promotion is to induce customers to buy the product on sale. This will, hopefully, draw in a profit large enough to cover the loss incurred with the price reduction. In some cases, it might even attract larger market shares when firsttime purchasers become loyal customers. Although this pricing strategy is often used in the real world, there is a drawback to it. It will only work if there are no similar products on sale. By having similar products on sale, customers face choices during purchase. Since we cannot accurately predict which product customers would choose, the potential profit and market share become unrealized. Because of this, the pricing strategy would actually produce less sales profit than before and render to be useless.
The companies understand this, and thus, they have never had any competing products go on sale simultaneously. This fits our logical assumptions above, but it is interesting to see that we have reached such an equilibrium that seemed so counterintuitive at first. This raises questions as to how the companies have reached such a decision. In fact, in the discussions following the article, one reader pointed out that this might be price collusion, in which the players cooperates and mutually agrees on a certain pricing strategy.
Although that might be true, but can it just simply be that letting one company promote at a time is a Nash equilibrium itself? We can examine this more closely by illustrating this game in a payoff matrix. Let’s define Coke and Pepsi as the two players with strategies of Promote (P) and Not Promote (NP). When the two companies both have a promotion offer, their pricing strategies would become useless as they both end up with less profits. Thus, each of the players will receive a payoff of 2. However, when there is only one company promoting, the promoting company will draw a larger market share and gain a payoff of 2. While the nonpromoting company loses some of its market share, and obtains a payoff of 1. In the case in which both company do not promote, nothing happens to their profit. Therefore, they would each receive a payoff of 0. Combining all these together, our payoff matrix would look like the following:

P 
NP 
P 
(2, 2) 
(2, 1) 
NP 
(1, 2) 
(0, 0) 
To find the Nash equilibrium, we first assume that player 1 plays P. The best strategy for player 2 would then be NP, since that gives a higher payoff of 1 than 2. In the case that player 1 plays NP, the best strategy for player 2 would be P, since that will give a higher payoff of 2 than 0. Because the two players have symmetric payoffs, we can use the same reasoning for player 1. We then arrive at the two Nash equilibria of this game – (NP, P) and (P, NP). This is exactly what we observe in the real world. Thus, using game theory, we can propose that instead of price collusion, the market naturally push the companies to arrive at these strategies as best responses to each other.