Utility Function

#Math

$\displaystyle \Delta \mathbb{E}[U_{X}]=\sum_{x = 1}^{\infty}p_{X}(x)[\ln(w+x-c)-\ln w]$

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

$\displaystyle \displaystyle \Delta \mathbb{E}[U_{X}]=\sum_{x = 1}^{\infty} \frac{1}{2^{k}}[\ln(w+2^{k}-c)-\ln w]x$