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Suppose someone offers you the following gamble:

You pay $7 and toss a coin. If the coin comes up heads, he pays you $10, and if tails comes up, he pays you $5.

You in turn get the idea of offering another person a coin toss in which he pays you $7 and tosses another coin. You tell him that if heads comes up, you will pay him $9 and if tails comes up, you will pay him $5. You think you see an opportunity to earn an arbitrage profit by engaging in both transactions at the same time.

Why is this not an arbitrage opportunity and how could you make it one, assuming you could get two people to engage in these gambles?

Answer:

In my opinion, This gamble is an arbitrage opportunity and but author said this gamble is not an arbitrage opportunity.

How is that?

Author's Concept checking answer is given below:

(General Arbitrage) The coin tosses are clearly independent. Thus, in some cases, the first coin will come up heads and the second tails, resulting in your earning $10 and paying $5. In some cases, the first coin will come up tails and the second heads, resulting your earning $5 and paying $9. You could make this an arbitrage only be linking the outcomes so that the $10 and $9 payoffs and the two $5 payoffs occur on the same outcome. An obvious way would be to toss a single coin but that might not be practical.

You could possibly link the payoffs to some type of event with two outcomes, such as whether the stock market goes up or down in a given period of time. Of course,you would still have to find two parties who would accept the odds and payoffs.

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  • It also depends on the definition of arbitrage. If you take it to intend that you can extract value with no risk, then this does not fit the description because the other person tosses a different coin. So even if your expected value were to improve with having the second game (which it does not), since the coin is different, you could in theory lose both coin tosses and realize a loss. If the games were based on the same coin, then you could construct an arbitrage by making sure that whichever side comes up, your profits are still >= 0, i.e. you have no risk
    – Ant
    Commented Jan 30, 2022 at 13:14

2 Answers 2

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The tosses of the two coins (assuming fair coins) are entirely uncorrelated. The payout of one coin has no connection to the payout of the other, even if the odds are the same. Now, if you could get both of them to make the same bets simultaneously on a toss of a single coin, that would be an arbitrage opportunity: your two partners disagree about the price they are willing to pay for the identical opportunity.

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The author is correct, this is not an arbitrage opportunity. To understand why, you need to look at the expected value in each game.

In the first game, in which you pay the $7 to play, let’s look at the possible outcomes, assuming a fair coin toss. If you win, you gain $3, and if you lose, you lose $2. There is a 50% chance of each outcome, so the expected value is in your favor. To you, the expected value in this game is +$0.50.

In the second game, where someone else is paying you $7 to play, the possible outcomes are you gaining $2 if your player loses, or you losing $2 if your player wins. This is a completely fair game; the expected value is $0, both for you and your player.

By you playing the first game, the odds are already in your favor. But you have not gained anything by offering the second game to anyone.

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