It actually is an interesting game (in the game theory sense). I make no claims about whether or not it is a good business idea, or how people will actually play, but a quick and incomplete theoretical analysis follows.

First, some simplifications: * an infinite number of players * everyone agrees on the ordering (quality, utility, etc) of shirts * only look for symmetric equilibria (where everyone plays the same strategy)

Some notation:

Let there be N shirts (N=5 for this application). Order the shirts by their utility (that everyone agrees on). x1 < x2 < ... < xN. Let the utility from the bonus shirt be z, where z > xN.

The most simple strategy is for everyone to randomize and put equal probability on each shirt. If everyone plays according to that strategy, then the expected payoff for any given player is avg(x1, ... , xN) + z/N. But this can't be a Nash equilibrium because an individual player could deviate and choose xN with probability 1 and get payoff xN + z/N.

The (maybe not the only) Nash equilibrium is probabilities p1, p2, ..., pN such that sum(p1, p2, ..., pN)=1 and x1+p1z = x2+p2z = ... = xN+pNz. These probabilities make a player indifferent between choosing each shirt with certainty.

I haven't spent the time to see if there is a nice formula for the probabilities, but for the case of N=2, you get

p1=(x2-x1+z)/2z and p2=1-(x2-x1+z)/2z

This doesn't take into account that people have different tastes, or the sequential information that the star creates. The game gets very complicated with these added in.

Gamification is the method of adding carrot/stick mechanisms to actions in order to drive behavior; it's a trivial application of game theory, which is about asking how people make decisions in (very contrived) situations.