The Goblin Game

Let's call $P_1,...,P_n$ the players, $\alpha_1,...,\alpha_n$ the total life points they have, $b_1,...,b_n$ the amount of life points they're ready to bet.

Without loss of generality, let's suppose $\alpha_1 \leq...\leq\alpha_n$.

EDIT: this is based on a payoff which would be $-\frac{\alpha_i}{2}$ if $b_i$ is the minimum, $-b_i$ otherwise, which is not what is requested.

The pay-off here seems to be $-\frac{-\alpha_i-b_i}{2}$ if it is the minimum or $-b_i$ otherwise.

Q1/2: It feels to me that the Nash equilibrium is to bet a number of points $b_i$ between $\frac{\alpha_{i}}{2}$ and $\frac{\alpha_{i-1}}{2}$ with $\alpha_0=1$ (imagining that you can have decimal lives for simplicity).

It is obvious that $P_1$ cannot improve his situation from that strategy. If he bets less, he will be losing the same because his bet will be strictly lower, if he bets more, well, he is worse off.

Knowing this, $P_2$ will want to bet as little as possible but ensuring he's staying ahead of $P_1$ so betting:

• at least $\frac{\alpha_{2-1}}{2}+1$ ensures he is not the smallest (or if he is, his loss will be only 50% vs $P1$ strictly more than 50%).
• at most $\frac{\alpha_2}{2}$, otherwise he is taking a hit for no reason

By induction, everyone will play along those lines, and the final bet is $\min(\frac{\alpha_{i-1}}{2}+1,\frac{\alpha_i}{2})$.

If everybody starts at the same level, then everybody declares 20 in your example, and everybody's life is divided by 2 at each stage. Any smart ass who plays more will lose more, and who plays less will not change the game.

That's a classical example of a Nash equilibrium where the equilibrium is to bring everyone down with you.

You can also note that if you get an edge, people can't team up against you, you're perfectly immune to their actions, there is no cooperation possible.

Q3:

• The angel doesn't change anything as your pay-off is unchanged from above.

• The Emperion: conratulations, you won the game no matter what, because everyone else will be losing at least 1 at each turn.

• The mirror: doesn't give you an edge either in the game, but will push you above everyone else at the last round, making you win the tournament.