Game Theory: Price-Matching Guarantees (PMG)

What is a price-matching guarantee (PMG)?

This is common to TV viewers, billboard watchers and general advertisement watchers alike. It's a guarantee by one firm to match the price of another rival for the same product. For instance, in the country where I am living in now, this happens in electronics stores.

The question to ask is this: Is such a seemingly competitive guarantee, truly in the spirit of competition?
What would you think?
Without thinking twice, I would say - 'Oh yes, a sign of competition indeed!'
Upon careful examination, one may actually find that such a guarantee is anti-competitive (this has been around in industrial organisation literature for a while now). Why is that? Well, let us take an example, which I will reference from Dugar (2005 and 2007).

Let us assume 3 firms dominating a product in an economy - what economists call a 'triopoly' (note: monopoly = one seller dominates the market; duopoly = 2 sellers, etc). Assume also that the sellers can choose to price their product between $1 and $100.

Case 1
There is no PMG option. The firm that chooses the lowest price reaps all profits, and the remaining 2 get nothing. Hence, the Nash equilibrium for this problem would be the price vector {1,1,1}. The profits, when split evenly, yield 1/3 per seller.

Case 2
All 3 firms adopt a PMG strategy. Regardless of where the firms choose their prices to be in the range between 1 and 100, all firms will match the lowest price offer (and hence split the profits evenly). Hence, the Nash equlibrium can be a symmetric set of vectors anywhere between {1,1,1} (which is the same as a non-collusive, competitive outcome) and {100,100,100} (the 'best' collusive, monopoly pricing outcome).

There are several other assumptions that come along with this game, but for the moment I will simplify the analysis (if you wish to read more, refer to Dugar (2005) or Dugar (2007) or Dugar and Sorensen (2006)).
It is easy to see that the PMG strategy is in fact a device for tacit collusion between firms - which is in fact, anti-competitive!

Interesting, isn't it?