Game theory question about financial markets

Question

This is a recent quote from one of the outstanding bond portfolio managers:

First of all, for every buyer there is a seller. Therefore, in order for someone to sell their bonds and buy stocks means that someone has to be selling stocks. It is a zero sum game.

I was wondering if this really is a zero sum game.

EDIT: What follows are my own musings, but my question really pertains to the quote above - Sorry for any distraction.

I know little about game theory. I would think that in a commodity transaction, where there is an explicit counter-party, whatever the buyer makes, the seller loses. (Or conversely.)

But in the above-mentioned transaction, it seems that both parties can make money. Assume I sell my bonds to you and with the proceeds buy your stocks. My new stocks can go up, and at the same time your new bonds can go up.

Thanks

Answer 1

This is responding to your edit, and you're on the right track. When professional investors talk about investments, you have to know when they are talking about fundamental value versus price action.

What you say in your edit is about fundamental value: stocks represent ownership in a company that may be generating dividends and earnings, perhaps at rates better than the market thinks, and so could go up in the future. Bonds tend to pay fixed coupons, but could still go up in price if interest rates or perceived risk falls. Both these examples are most certainly not zero-sum.

But your portfolio manager was making a classic short-term price action statement. If I dump bonds now to buy stocks now, then in the very short term the demand for that stock goes up and the bond goes down, so you get a slight price increase on the stock and slight price decrease on the bond, which is "zero-sum". Or so the thinking goes. And yeah, some other guy had to sell his stock to the buyer, so "zero-sum" in the very short term.

I happen to think this is erroneous, but you need to be aware that professional investors talk like this all the time, assuming very short-term supply/demand thinking, which has a zero-sum flavor. But your observation about the fundamental long-term value is closer to the truth.

Answer 2

When I pay you \$100 for a share of stock, I believe that the stock is worth at least \$100 (or I wouldn't buy), and you believe that the stock is worth at most \$100 (or you wouldn't sell). It's possible both of us think we came to advantage in our personal utility functions. There is no intrinsic "value" to the share, only what you and I agree to pay.

Later, the company associated to the stock can be successful, and suddenly everyone thinks the share is worth more than \$100, but that is a consequence of a change as time marches on, not a consequence of our transaction.

Answer 3

In your example both parties would have made money even without the transaction. Assuming that the payoffs are in dollars rather than utils,* that particular game is constant-sum.

Zero–sum games are a specific example of constant sum games where the sum of each outcome is always zero. Such games are distributive, not integrative; the pie cannot be enlarged by good negotiation.

^Wikipedia

Since payoffs can always be normalized, constant sum games may be represented as (and are equivalent to) zero sum game in which the sum of all players' payoffs is always zero.

^{Game Theory .net}

Suppose that you own some bonds and some shares. If this weren't a constant-sum game, you could sell the bonds to yourself and, with the proceeds, buy the shares from yourself. Obviously, that wouldn't put you in any better position than before the self-dealing. (Except for, e.g., market manipulation and accounting fraud, but that's a whole different ball game.)

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Edit due to comment/changed question:

No, it's not (necessarily) a zero-sum game. The key point is that Game Theory typically deals with utility, not money. In that sense the "outstanding bond portfolio manager" is totally off the mark, if he means this in a game-theoretical sense. However, if you equate the two,* then yes, this is a constant-sum game, and therefore game-theoretically equivalent to a zero-sum game. Also note that games are sometimes part of bigger games. And so is this one.

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Looking up the quote on the Internet, I found that it is merely supporting the non-happening of the "great rotation": people leaving one asset class en masse, to enter another one. He simply means that, at any given time, the number of shares (and the outstanding notional in debt) is fixed, therefore any trade cannot change the ratio between them. The overall net exposure to an asset class cannot be changed instantaneously. If used this way, it has nothing to do with game theory.

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^{*This is an assumption that typically doesn't hold in portfolio theory. Game theorists would (and did) say: "To avoid difficulties, assume dollar payoffs and linear utilities."}

Answer 4

No. Otherwise nobody would trade. You can explain trades either by some risk-diversification story or by both having the subjective belief that they would one-up the other. Such subjective beliefs have been analyzed in game theory in the paper Subjectivity and Correlation in Randomized Strategies by Robert Aumann. But even subjective beliefs don't really help in zero-sum games as shown in Proposition 5.1 in that paper.

Answer 5

Yes and No.

No, because 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 trade.

Yes, if you allow irrational traders to exist. One hypothesis is , other than arbitrageurs, there also exists noise traders, who are subject to systematic bias, buy and sell randomly but in unison. Their existence, associated with their irrational and unpredictable behaviour pose extra risk to rational speculators. In J B De Long et al(1991), contrary to conventional wisdom, traders who commits systematic errors may not necessarily be driven out of market.