# Expected value and optimal strategy for red/blue game

Firstly please excuse my ignorance if I'm posting this to the wrong exchange site. If this doesn't belong here let me know and I'll move it.

Now as for my question, today during a short course that I am studying our instructor gave us a game they call RED/BLUE.

My gut instinct instantly told me that there is both an optimal strategy for how to play this game and that each one of the decisions can be assigned an expected value over x games but I am completely mathematically illiterate and hope that someone could please tell me if:

A. Am I correct? Is there an optimal strategy for this game?

B. How can one calculate the expected value of the decisions in a game like this? (If that's even the correct term)

I'd really appreciate a layman's explanation.

P.S. Obviously I understand that the inclination is to pick red every-time assuming the other group will remain honest but surely it isn't that simple?

This is effectively an iterated prisoners' dilemma. If it was a single round then you would do better to play Blue rather than Red, no matter which colour your opponent selects in the same round, but if there is a series of repeated rounds then trust between the two sides can lead to both choosing Red and ending up with positive scores so long as the trust is maintained.

There is no provably optimal strategy which leads to positive outcomes that does not involve communication with the other team. The assumption is that such commmunication only takes place within the game and any punishments can only be delivered in the game.

The problem is that there is no reason to be trusting in the final round if it is known to be the final round, and this break-down of trust feeds back through the earlier round if both players are game-theoretic rational. Despite this, some trusting strategies with retaliation can evolve in a population with other similar strategies present.

• This is not a true iterated prisoner's dilemma, since all choices have to be made before the games are played. Commented Jan 28, 2014 at 13:59
• @Arthur "Colours are chosen immediately prior to each round and you have three minutes between rounds". At the very least this means you can decide a complete strategy to deal with a finite number of possible earlier rounds. In practice, the game is designed to allow thought between each round and presumably discussion within each team. Commented Jan 28, 2014 at 14:01
• Oh, I misread, thought that you handed in a list of your choices to the gamemaster who then calculates the results. As for strategies, tests (there have been competitions where you send in your strategy, and then they pitch yours against all the other strategies randomly, but with a high probability of meeting the same opponent in consecutive rounds) have shown that "retaliation" (picking what your opponent picked the last round) does pretty good compared to most other strategies, especially for being so simple. But again, this is whithout knowing when it's going to end. Commented Jan 28, 2014 at 14:03

If all players are rational players and we assume that one of the moves is optimal then it is rational to assume that both players arrive at the same conclusion by the same rational methods. In such a situation, it can be assumed that either both play RED because RED is rationally better or both play BLUE because BLUE is rationally better. Since the first give both $+3$ and the latter gives both $-3$, only RED can be the rationally best strategy under these assumptions. The problem that remains is:

• Is my opponent as rational as that?
• Am I smarter than my opponent and can outsmart him (at least in the last round or rounds)?
• May he think he's smarter and is likely to try and outsmart me in the last rounds?
• Should I think that he thinks that I think that he fears that I might be afraid of him thinking that I believe he might want to outsmart me? ...