Is this game expectation correct? Discovery Kids' "Endurance 2"

Factual Questions
1

I saw the last episode of Discovery Kids’ Endurance 2, a young-teen version of Survivor. Watching them play the final game, I kept thinking, “Stupid kid! You’re doing it all wrong!” Then I realized that I may be the stupid one here.

To minimize hassle, here is a siplified version of the game:

[ul][li]There are two players, 1 and 2[/li][li]There are a number of boxes and we let n represent their number[/li][li]One box is randomly chosen to contain a ball, the players have no knowledge of which box the ball is in[/li][li]Player 1 bets one unit per box, on as many boxes as she wishes, as long as she bets on at least one box and no more than n-1 boxes[/li][li]Player 2 must bet one unit each on the remaining boxes (that player 1 didn’t bet on)[/li][li]The winner is the player who bets on the box with the ball in it[/li][li]The winner receives all units bet; so payoff P=n[/li][/ul]

So, player 1 increases her chances of success if she bets on more boxes, but she decreases the payoff as she does so. That second part was the part I wasn’t clueing in on.

So, what is player 1’s expected gain, E[P]? It’s the odds that she bet on the correct box times the number of units she gains, plus the odds that she bet on the wrong box times the number of units she loses. Let’s call her bet “A” and player 2’s bet “B”.

(Since I don’t know how to code fractions, please bear with me.)

Her expcected gain is then:

E[P] = (A/n)•B + (B/n)•(-A)

    = (AB/n) - (AB/n)

    = 0.

So, player 1 expects to gain zero units no matter how many she bets.

Of course, she will increase her risk by increasing the size of her bets, since the variance will grow the more she bets, so that in a repeated game she may want to play a minimum risk strategy. If someone wants to analyze this and show that my intuition is way off or spot on, feel free to do so. But what I’m mainly concerned with is whether I have the right result for player 1’s expected gain on each bet.

So, do I?

Thanks.

2

I think you have it just right. The expected outcome is ‘0’. This is a zero sum game. If played enough times, both players should break even, unless there are additional details you haven’t mentioned.