# Is risk-taking a zero-sum game?

I learnt that trading is a zero-sum game since profits take out losses while holding your assets may not always be a zero-sum game when time passes and everybody's assets theoretically can grow in value.

But is risk-taking also a zero-sum game? Risk as measured by volatility (or S.D.) like taking risk in a financial market? Is the problem of risk-taking reducible to the problem of trading?

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This question invites long answers, so let me just give a few thoughts.

Preferences: suppose you have risk averse Linda and risk loving John. Linda owns a house, and she is afraid it might burn down. Now it is possible to make both better off by designing insurance: Linda pays insurance premium $x$ per month to John, and in case her house burns down, John buys her a new one. Thus, the risk is shifted from Linda to John. John is happy to receive it, because he is risk loving. Linda is happy to get rid of it, because she is risk averse. Hence: not a zero-sum game if there is heterogeneity in risk preferences.

Financial markets: consider futures contracts, which are basically bets on future prices. You are right that, viewed in isolation, when you buy such a contract from someone, either you lose or he does. But suppose you buy that futures contract to insure your business against falling crop prices (you are a farmer) - that is, you bet on falling prices. In that case, you actually want to lose the bet - it is just a security measure against the worst case. Hence, the mere possiblity of trading those securities can improve the welfare of all, because it too acts as insurance and can allocate risk from those who don't want it to those who do (for a price, of course).

Since you added the game theory tag: I am sure it depends on the game whether risk trading can be a zero-sum game or not. In a game where all agents have the same risk preferences, someone has to take the hit, but nobody wants to. But as outlined above, with sufficiently diverse risk preferences there may be arrangements to satisfy all agents.

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Zero-sum game is a bit too strong. A more general and realistic assumption is that the no trade point is on the pareto surface. One trader's gain must be at the expense of another.

One important and closely related question to poster's, is, from a game theoretic perspective, why trades occur in finanial market, or its weaker version, why trades in finantial market of a narrow sense occur in such a high volumn, even taken diversity of risk preferences and asymmetric information into account.

As pointed out in Nameless's answer, in basic Lucas tree model, assuming homogeneous traders, there's no room for trade. The market price of asset is totally determined by the conditions that guarentee a representative trader has no incentive to buy or sell a piece of an asset.

Even worse, Milgrom and Stokey(1982) shows that ex ante pareto optimality of no trade point is incompatible with common knowledge at some state that no trader is worse off and at least one of them is strictly better off after a non-zero net trade. Noticeably, the result of no trade is obtained by only assuming traders are all risk-averse.

In conclusion, there's some element of truth to see trade in some market, say, stock market, as a zero-sum game, despite of the existence of a more realistic alternative assumption. Diversity of risk preferences is far from the whole story of why trades occur. As an attempt to explain it, noise traders has been hypothesized by some behavioral economists.

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