Expected Value and Fair Pricing in a Two-Draw Lottery

Summary

The document asks how to price a simple lottery in which a player draws two balls without replacement from a bag containing two red, two green, and one white ball. Drawing the white ball pays a fixed amount; the player pays an entry price. The author calculates the expected net payoff and asks whether setting it to zero gives the fair price, and how that relates to martingales and option pricing.

The calculation illustrates a basic expected-value definition of fairness: choose the entry price so the player’s average net payoff is zero under the stated probabilities. The document does not provide a full answer or develop an option-pricing model. Its question also leaves room to distinguish this actuarial fair price from prices that reflect risk preferences, market pricing, or replication under a pricing measure. The example is a simple finite-outcome game, not a trading strategy or empirical analysis.

Key ideas

• A fair entry price under the expected-value criterion makes the expected net payoff zero.
• The payout probability follows from drawing without replacement from the specified bag.
• A zero expected payoff does not by itself address risk preferences or market pricing.
• The document poses a connection to martingale pricing but does not resolve it.

Tags

# simple game - fair value


# simple game - fair value












Suppose a person A has the following game:

- there are 2 red balls, 2 green balls and 1 white ball in a bag

- you take 1 ball (don't put it again in the bag) and then a second ball

- if you take the white ball, you win 10$.

The question is : what is the price of this game ? Suppose this is $c$ , then one can find that : $E[X] = 4-c$, with $X=10-c$ or $X=-c$. So in order to have a fair game : $c=4$. Is it the good answer to find the $fair$ price of this game ? I think there is the notion of martingale for the "fair" price. Fair price would mean that A, and the person who plays have equal chances in this game. And because it's "fair" we would have $E[Wealth_{t}|F_{s}]=Wealth_{s}$ with $Wealth=X$ actually. So we have to write $E[X]=0$ in order to find c, right ? I would like an intuitive answer (I know the numerical part is right ^^) by using some concepts of the option pricing theory (?).

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.