Hi everyone,

there is currently a betting game going on among colleagues and I was wondering what the optimal strategy would be for the final game if the 2nd best player could only overtake the best player in the last game if he bets on the winning team while the currently best player bets on the losing team.

There is one favorite for the final game, e.g. France is favorite over Croatia, maybe 60/40. The players do not know each other's bets before the game starts.

What is the optimal strategy for both players, if there is one?

Player1 would of course prefer to just pick the same side as Player2. Player2 probably assumes that Player1 chooses the favorite and therefore would pick the underdog, but if Player1 knows that, then it would be best for him to pick the underdog as well...

Can this be put into context with Nash Equilibrium, Game Theory, Poker Analogy?

Thanks!

Yep. The easiest way to do this (although I have never seen anybody but me do these type of problems this way) is to assume that the opponent always picks the favorite, calculate your EV and do the same if he always picks the underdog and see what strategy gives you identical EVs. This always gives you the answer regardless of his strategy . Do you see why?

In this case assume you are ahead and want to know how often to pick the 40% shot (call that x) and how often to pick the 60% (1-x).

If he always picks the underdog you chances of winning the contest are x(instant win) plus (1-x) times 60%.

If he always picks the favorite your chances are x times 40% plus (1-x) (instant win).

x+ .60 -.60x = .40x +1-x

.40x +.60 = 1-.60x

x = .40

So the answer is that you pick the underdog 40% of the time and win the contest 76% of the time.

I will leave it to you to solve the general case.

Hi, thanks for your reply!

I think I got it, just a few follow-up questions:

So for my opponent it would be optimal to choose x (the underdog) 60% of the time?

EV if I pick x = EV if I Pick (1-x)

0x+(1-x).60=(1-x)0+.40x

.60-.60x=.40x

x=.60

If one of the players deviates from these strategies, the other player could bet a 100% on the best choice for him and increase their chances of winning, correct?
If he knows I always bet on the favorite his chances would increase to 40% and if he thinks I bet more than optimally on the favorite, say 80% of the time, he could increase his chances to 32%.

Therefore I should bet on the favorite that % of time as they are favorite to win and my opponent should bet on the favorite that % of the time as they are to lose?

If there is no favorite (50/50) I should flip a coin and I am 75% to win which is nearly the same as if its 60/40. If its a clearer favorite like 80/20 my chances would be 84%.

Is this all mostly correct or are there any mistakes in my conclusions?

Thanks!

Looks right.