Utility Function

$\Delta \mathbb{E}[U_X]=\sum _{x=1}^{\infty }{p}_X(x)[\ln (w+x-c)-\ln w]$

• $\Delta \mathbb{E}[U_X]$ is the change in the expected value of the utility of a random variable $X$
  • If greater than 0, a rational person would take this bet. Otherwise, they wouldn’t
• $X$ is the random variable representing winnings in dollars for this particular bet without factoring cost
• $x$ is the winnings in dollars from the bet for a certain event
• $p_X(x)$ is the probability of winning $x$ dollars
• $w$ is the wealth of the player
• $c$ is the cost to play the game

$\Delta \mathbb{E}[U_X]=\sum _{x=1}^{\infty }\frac{1}{2^k}[\ln (w+2^k-c)-\ln w]x$