05.05.06
Stabilizing Collusive Optimums
At the end of my post about Nash Equilibriums I teased that I may discuss how collusive optimums could possibly be stabilized. Well I decided I would all of my loyal readers… (NOTE: this discussion assumes a multi shot game, which I haven’t really discussed yet. I will though, eventually 
)
First, I will talk about why collusive optimums are typically unstable in the first place. Then I will illustrate one of my favorite examples of how to make them rock solid
Collusive optimums are typically unstable because there is a LARGE incentive to cheat, and we all know that everyone is a cheater… Notice, in our example from before, that if Sav-on and Walgreen’s got together and decided that they would both price HIGH that both parties would be better off than in the Nash Equilibrium. However, also notice that if, instead, either party (but not both) agreed to the higher pricing and then cheated by setting a low price they would actually make out better. This is the incentive to cheat.
How do we overcome this? Well first there is a problem of credibility. Walgreen’s can tell Sav-On that if they lower their price we will immediately lower ours, but there is a problem with this threat; it’s not credible. Walgreen’s would MUCH rather forgive Sav-On for being so scandalous and reinstate the high price agreement because they would make more money that way. This often happens in collusive oligopolies (or cartels), such as OPEC. They get together and decide on a production ceiling that each country will not produce beyond. In a free market this sets price… Inevitably what happens, though, is that each country has an incentive to produce just a liiitttle more . That way they enjoy the extra profits from the added volume at the expense of their associates. Then the other countries come and slap their wrist and sit down to agree to another farcical production ceiling.
Ok… so we need a credible threat. Threats could be made credible by prior history of enforcement. For example, the airline industry has time and time again instituted a fare war, incurring massive losses, in order to dissuade firms from lowering prices. Another credible threat would be an advertised promise to follow through with a price reduction. This is the all to familiar ‘lowest price guarantee’ See you ignorant consumer… you thought that was actually something that would be in your benefit didn’t you? Oh Contraire! Empirical evidence shows that prices, in fact, rise when these policies are put in place. Let’s use our Sav-On example to illustrate…
If both Sav-On and Walgreen’s announce that they will match any LOW move made by its opponent prior to making their move it then makes it possible for both players to price HIGH. It would now be disadvantageous for either firm to price LOW because it KNOWS its opponent will price LOW as well. We’ve now removed the incentive to cheat by making a credible threat and in so doing we’ve created a stable collusive equilibrium. We’ve also modified the Normal form of our game, by removing both the (HIGH,LOW) and (LOW,HIGH) states we are left with only (HIGH,HIGH) and (LOW,LOW). Faced with only these two states both companies will choose to price HIGH.
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