The original ICM model operates as follows: • Every player's chance of finishing 1st is proportional to the player's chip count. • Otherwise, if player finishes 1st, then player finishes 2nd with probability \mathbb{P}\left[X_i=2\mid X_k=1\right]=\frac{x_i}{1-x_k} • Likewise, if players , ..., finish (respectively) 1st, ..., st, then player finishes jth with probability \mathbb{P}\left[X_i=j\mid X_{m_z}=z\quad(1\leq z • The
joint distribution of the players' final rankings is then the product of these
conditional probabilities. • The expected payout is the payoff-weighted sum of these joint probabilities across all possible rankings of the players. For example, suppose players A, B, and C have (respectively) 50%, 30%, and 20% of the tournament chips. The 1st-place payout is 70 units and the 2nd-place payout 30 units. Then \mathbb{P}[A=1,B=2,C=3]=0.5\cdot\frac{0.3}{1-0.5}=0.3\mathbb{P}[A=1,C=2,B=3]=0.5\cdot\frac{0.2}{1-0.5}=0.2\mathbb{P}[B=1,A=2,C=3]=0.3\cdot\frac{0.5}{1-0.3}\approx0.21\mathbb{P}[B=1,A=3,C=2]=0.3\cdot\frac{0.2}{1-0.3}\approx0.09\mathbb{P}[C=1,A=2,B=3]=0.2\cdot\frac{0.5}{1-0.2}\approx0.13\mathbb{P}[C=1,A=3,B=2]=0.2\cdot\frac{0.3}{1-0.2}\approx0.08\mathrm{ICM}(A)=70(0.3+0.2)+30(0.21\cdots+0.13\cdots)\approx45\approx90\%\mathrm{ICM}(B)=70(0.21\cdots+0.09\cdots)+30(0.3+0.08\cdots)\approx32\approx110\%\mathrm{ICM}(C)=70(0.13\cdots+0.08\cdots)+30(0.2+0.09\cdots)\approx22\approx110%where the percentages describe a player's expected payout relative to their current stack. == Comparison to gambler's ruin ==