Price wars are the rule, not the exception.

Chain Store Paradox

The chain store paradox is a simple game in an extensive form i.e. the players understand the potential strategies of each player and as a result, their own best course of action to maximize their payoffs which produces an inconsistency between game theoretical reasoning and plausible human behavior. Well-informed players must be expected to disobey game-theoretical recommendations. Before moving ahead, let’s first the meaning of price war.

A price war is a competition among the competitors of the business in lowering the price of their products to gain an advantage over their competitors in price and to capture a greater market share.

Suppose a chain store is challenged in town 1 with the first rival. Now, it has two options, either to go for price war or acquiesce. After observing this decision, the second rival of this chain store in another town 2 has to decide whether to enter into the market or quit. And then after that, the Chain Store in town 2 has to decide again whether to go for price war or acquiesce with the rival there (town 2). Both chain stores belong to the same company, therefore, having the same policy about marketing.

The Chain Store earns 0 each time it goes to war, 2 each time it acquiesces and 5 if the second challenger quits. The second rival earns -2 if the chain store goes for war, 0 if it quits, 2 if the chain store goes for acquiescing.

Here only the second rival gets the chance to choose to enter the market or quit seeing the decision made by the Chain Store in town 1. So whenever we would be talking about the rival choosing to enter the market (or not) or about points earned, it would be the second rival.

The Game Tree:

(Fig. 1) The game tree when Acq. is more preferable than war for the chain store

Here, if Chain Store at first chooses to acquiesce in the first city with the first rival, then it achieves 2 points and then if the second rival decides to quit the market against the chain store in town 2, the points earned by the Chain Store would be 2+5= 7 and 0 for the rival as it quits.

Or if the rival challenges and enters the market in town 2, then Chain Store in town 2 has to decide whether it would go for price war or acquiesce if it goes for acquiesces, the points earned by the Chain Store would be 2+2= 4 as chain store in town1 also went for acquiesce and 2 points for the rival. And if it(chain store in town 2) goes for war, the points earned would be 2+0= 2 for the chain store and -2 for the rival.

Now if initially, the Chain Store in town 1 goes for war with the first rival, then it gains 0 points and then if the second rival decides to quit the market against the chain store in town 2, the points earned by the Chain Store would be 0+5= 5 and 0 for the rival as it quits. Or if the rival challenges and enters the market in town 2, then the Chain Store in town 2 has to decide whether it would go for price war or acquiesces, if it goes for acquiesces, the points earned by the Chain Store would be 0+2= 2 as the chain store in town1 went for war and 2 points (as the chain store in town2 acquiesce) for the rival. And if the chain store in town 2 goes for war, the points earned would be 0+0= 0 for the chain store and -2 for the rival.

We can find subgame perfect Nash equilibrium, a Nash equilibrium of every subgame of the original game by using backward induction which is to reasoning backwards in time, from the end of a problem or situation. The Nash equilibrium is a decision-making theorem within the game theory that states a player can achieve the desired outcome by not deviating from their initial strategy. In the Nash equilibrium, each player’s strategy is optimal when considering the decisions of other players.

Lowest Level of Tree:

So by using backward induction, here the chain store in town2 would definitely choose to acquiesce as points earned by the Chain Store would be greater(4>2 and 2>0) in both of the cases or subtrees if chain store in town1 initially goes for war or acquiesces.

Choosing to acquiesce will always fetch the chain store greater points, irrespective of the initial choice.

(Fig. 2) The Chain Store decides at the lowest level, whether to go for war or acquisition

One More Level Up:

The “war” part of the tree is truncated as we have just seen that acquiesce fetch more points for the Chain Store. Now going one level above, the second rival would choose to challenge and enter the market as points earned by the rival would be greater(2>0 and 2>0) in both the subtrees.

Note: Don’t see the “war” part as it got rejected by the Chain Store.

(Fig. 3) Rival decides whether to challenge the chain store or to quit

Final Game Tree:

Now, the “quit” part of the tree is truncated. Now going one more level to the top, the chain store would be choosing to acquiesce as points earned by it would be greater(4 > 2).

(Fig. 4) Final game tree where acquiesce is more preferable than war for the chain store.

The equilibrium here is that the chain store chooses for acquiesces all the way through and the rival enters the market. However, this generally doesn’t happen in real life! In reality, the chain store would try to signal strength and go for war indicating that it would go for war again if the second rival entered the market. This would be done in order to avoid the second rival from entering the market. So if the chain store succeeded and the rival does not enter the market, then it would get a payoff of 5 (0+5= 5, 0 as it goes for war initially).

The problem of calling it a paradox is due to the fact we are trying to map a complete information model onto an incomplete information environment. So let’s make a complete information environment.

Complete Information Theory:

The assumption of complete information: that “every player knows the payoffs and strategies available to other players,” where the word “payoff” is descriptive of behavior — what the player is trying to maximize. If in the first town, the Chain Store would have gone to war and got itself into losses just to avoid the second rival entering the market, it would have made its losses through the town2.

Here, we have an assumption whether the Chain Store would go for war or acquiesces. Later, we would make this up to complete information by coming up with a belief.

For now, Considering that the price war is more powerful in the long run, then let’s see, what happens!

Game Tree when War is preferable:

Suppose the Chain Store earns 4 each time it goes to war, 2 each time it acquiesces and 5 if the second challenger quits. The second rival earns -2 if the chain store goes for war, 0 if it quits, 2 if the chain store goes for acquiescing.

(Fig. 5) Game tree when war is more preferable than acq. for the chain store.

Now, this is a game tree when war is a better option for the Chain Store. Here also, by using backward induction, you can think that equilibrium is when the Chain Store goes for the war initially and the rival quits.

As at the lowest level, the Chain Store would go for war definitely as points earned would be greater(6> 4 and 8> 6). Now Acq. got rejected by the Chain Store. Then Rival would like to quit as it would be better for it(0> -2 in both cases). Now on the top level, Chain Store would like to go for war as points would be greater(9> 7).

So we have two conditions when acq proves to be a better option for the Chain Store and when war is better.

Now, suppose the rival is not sure that the chain store would go for a price war or not initially. Let the rival make a belief that there is a 10% chance that the chain store will not go for war and a 90% chance that it would. The belief made by the rival on the basis of the market reality. Now, the game tree looks like this with all probable payoffs.

(Fig. 6) Overall analysis on both the previous game tree on the basis of belief of the second rival.

Let’s now analyse the strong type of chain store for a moment. We can observe that it goes for “war”, the payoff i.e, 8 and 9 is more than the payoff it gets when it goes for “acq”.

So clearly, if the strong type chain store (like a big company) goes for a price war, it would get the maximum payoff. And for the weak type like a small company, it depends whether it would succeed in stopping the rival from entering or not.

The Chain Store Paradox is everywhere where the competition takes place — politics, markets, business, defence, etc. In politics, choosing a constituency candidate to leave the opposing party to decide to change their candidates or not is a good example. Thus, if you find yourself in a chain store paradox, don’t forget to make a game tree and strategically predict the outcome!

REFERENCES:

• Reinhard Selten(1978). “The chain store paradox”. https://link.springer.com/article/10.1007/BF00131770#citeas
• Crushing the competition- at any price, Financial times https://www.ft.com/content/b7f4a4e6-3ec1-11e4-adef-00144feabdc0
• Aumann, R. J., ‘Mixed and Behavior Strategies in Infinite Extensive Games’ http://www.ma.huji.ac.il/~raumann/pdf/Mixed%20and%20Behavior.pdf
• Game Theory 101:The Complete Textbook book by William Spaniel
• https://en.wikipedia.org/wiki/Chainstore_paradox#The_chain_store_game