Log Utility and the Maximum Entry Fee for a Gamble
Summary
The document explains how to find the maximum entry fee a log-utility consumer would pay for a coin-flip gamble. The example gives a starting wealth of 100, with a payoff of 20 on heads and zero on tails. Because the fee is not specified in the question, a respondent supplies the needed criterion: choose the fee that makes expected utility after paying equal to utility at starting wealth. This is the consumer’s reservation price under that rule.
The calculation takes the probability-weighted average of log wealth in the two outcomes and sets it equal to the log of initial wealth. The document reports an approximate fee of 9.5 for the initial example and about 9.9 when wealth is 1,000. These figures illustrate that the richer consumer is less sensitive to the same potential loss in wealth. The result depends on the assumed log utility, equal coin-flip probabilities, and the indifference criterion; the discussion does not model other preferences or constraints.
Key ideas
- The maximum entry fee is defined by indifference between playing and keeping the initial wealth.
- Expected utility is the probability-weighted average of log wealth across outcomes.
- The example uses a coin flip with a positive payoff on heads and no payoff on tails.
- The reported reservation fee rises slightly when starting wealth increases.
- The conclusion depends on the log-utility assumption and the stated payoff structure.
Tags
Full text
# What is the most amount of money the consumer would be willing to pay to play take this gamble?
# What is the most amount of money the consumer would be willing to pay to play take this gamble?
Suppose a consumer has log-utility over wealth, defined by $u(W) = \ln(W)$. Suppose this consumer has $100$, and is considering taking a gamble in which the consumer flips a coin, and gets $20$ she flips heads and $0$ if she flips tails. What is the most amount of money the consumer would be willing to pay to play take this gamble? What if the consumer has $1, 000$ dollars of wealth?
I feel like this question is incomplete since we do not know how much the person need to give in order to play the game. Do we know that information in order to solve this problem?
If no, could anyone tell me the direction and piece of information needed to solve for this question?
## Answer by user1157 (score 1)
https://quant.stackexchange.com/a/10258
This is a basic utility exercise. I would guess the additional assumption you are missing to solve the exercise is that the player would be willing to accept the fair game but nothing worse.
To make this a fair game, the maximal amount which could be paid to enter the game, would result in zero expected loss of utility (nobody would accept anything worse). In other words, the expected utility must match the utility of the starting wealth $W_0$: $$ \mathrm{E}[\ln(W)]=\ln(W_0)$$ Since this game has only two outcomes you can easily compute the expectation from the tree of possible outcomes
```
/ Head: 100 + 20 - entry fee
/ (50%)
/
100
\
\ (50%)
\ Tail: 100 + 0 - entry fee
```
From this we can compute the expectation directly: $$\mathrm{E}[\ln(W)] = 0.5\ln(120-x) + 0.5\ln(100-x)=\ln(100)$$
The result is approximately 9.5 (you can check here), and if you repeat with a starting wealth of 1000, the result is approx. 9.9, since the richer person doesn't care as much about loosing a few bucks as the poorer guy does.
## Answer by Jane Bond (score -1)
https://quant.stackexchange.com/a/10256
If there is not a entry fee, I cannot lose money for at worst, I win 0USD.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.