Game Theory

Equity prices have vaulted to within touching distance of all-time highs, and the upturn in investors’ fortunes has pushed valuation ratios to levels that have preceded protracted periods of poor stock market performance in the past.  The widespread belief that stock market returns mean revert over long horizons means that it could well be possible to use the information contained in high valuation ratios to time the market, and capture the favorable combination of lower risk and higher returns.

An interesting paper authored by Elroy Dimson, Paul Marsh, and Mike Staunton of the London Business School, and recently published in the ‘Credit Suisse Global Investment Returns Yearbook,’ casts doubt on this view.  The academics assess the predictive ability of a cyclically-adjusted price-dividend ratio – the ratio of the current real index level to the average of the preceding ten years’ real dividends – across a variety of world stock markets, and conclude that, “we learn far less from valuation ratios about how to make profits in the future than about how we might have profited in the past.”

The choice of valuation metric appears reasonable, since dividend payments, unlike earnings, cannot be manipulated, and often reflect a company’s own view of its long-term earnings power.  However, there is no theoretical reason as to why a cyclically-adjusted dividend-price ratio should mean-revert, since the higher multiple might simply reflect substantive changes in the percentage of earnings that companies decide to pay to shareholders.

It is important to appreciate that stock market value is made up of both current dividends and expectations for future growth.  The pace at which dividends grow in the future depends on the percentage of earnings that a company distributes to its owners, and the rate at which retained earnings are reinvested in the business.  In other words, a low dividend yield might simply reflect a lower payout ratio, and higher expectations of future growth.

Historical data for the US demonstrates that the corporate sector’s payout ratio has been in secular decline for decades.  The ten-year average payout ratio dropped from a peak of almost ninety per cent in 1940, when expectations for future growth were virtually non-existent, to below forty per cent in 2007, when expectations for uninterrupted growth for the indefinite future held sway.  Long-term differences in payout policy means that it is impossible to identify a mean around which the cyclically-adjusted dividend-price ratio might oscillate.

Financial theory suggests that we should be able to observe a negative relationship between corporations’ payout ratios and subsequent growth rates in earnings and dividends.  In other words, higher growth rates would be expected to follow lower payout ratios and vice versa, but if this expectation is frustrated, then the dividend-price ratio might retain some predictive ability, as disappointing growth outcomes are reflected in lower share values.

The historical evidence in both the UK and the US reveals that low payout ratios have typically been followed by surprisingly low real growth rates over subsequent ten-year periods, and not the high rates of expansion that might have been expected at the outset.  This surprising outcome suggests that the corporate sector is either over-investing, or that  competitive markets quickly erode excess returns, or that low payout ratios reflect management’s intention to signal lower future growth to shareholders.

In light of the above, the dividend-price ratio does retain some predictive ability regarding future real returns, but it is still not possible to say what level is indicative of fair value.  As a result, it would be wise to replace the cyclically-adjusted dividend-price multiple with a valuation metric that rests on sounder theoretical footing.

In this regard, the Q-ratio, developed by the late Nobel laureate James Tobin in 1969, is a natural choice.  This metric measures the market value of equity relative to its replacement cost, and a fundamental relationship should exist between the market value and replacement cost; corporations should be valued at their cost of creation in the long-run, and as a result, the multiple should hover around unity given rational expectations.

The “law of one price” or “build-or-buy” arbitrage should ensure that the relationship holds over long horizons.  A ratio above unity implies that it is cheaper to invest in new capital rather than buy existing capital, while a figure below unity suggests the opposite.  The historical data confirms that the Q-ratio does indeed demonstrate mean-reverting properties, and importantly, the analysis reveals that the adjustment takes place through a change in real share prices rather than changes in the capital stock.  In other words, Tobin’s Q can be used to predict long-term real returns.

Unfortunately for equity investors, the current value of Tobin’s Q is almost forty per cent above its long-term mean – a level that has rarely been exceeded in the past.  The ratio’s elevated level, in tandem with its mean-reverting properties, does not mean a catastrophic decline is imminent, but it does suggest that disappointing real returns are virtually assured over long horizons.

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Be sure to read the full risk disclosure before trading Forex. Please note that Forex trading involves significant risk of loss. It is not suitable for all investors and you should make sure you understand the risks involved before trading. Performance, strategies and charts shown are not necessarily predictive of any particular result. And, as always, past performance is no indication of future results. Investor returns may vary from Trade Leader returns based on slippage, fees, broker spreads, volatility or other market conditions.

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One of the games which is repeatedly used as an example in game theory is called the Prisoner’s Dilemma. In this game, two prisoners are caught and they are each given the choice to cooperate with each other or turn on each other. This game has been used to simulate various real life scenarios including many economical situations and in essence describe any un-trusted transaction between people.

In 1984, Robert Axelrod conducted an experiment in which he asked scientists to play in a tournament of repetitive prisoners dilemma with one another – by creating the repetition of the game Axelrod made it possible for people to adapt their strategies and also created a memory in the game such that each player plays the next iteration with the knowledge of the strategy his opponent played in the previous iteration.

The winning strategy was found to be "tit-for-tat", which leads to a balance of cooperation between the two prisoners.

Axelrod then conducted another tournament where he disclosed the winning strategy. The surprising thing he found was that even though everyone knew the winning strategy, it still remained the winning strategy. So collaboration on the strategy did not affect the game in any way.

Another variation of the prisoner’s dilemma is called an assurance game and is most commonly known as “Hunt a Stag.” In this game, two hunters can choose to collaborate and hunt a Stag (large and tasty adult deer) or split and hunt a rabbit alone (smaller but just as tasty).  The difference between this game and the Prisoner’s Dilemma is that players are allowed to communicate and choose to collaborate.

So creating a proper mechanism for communication and trust between players is essential for collaboration.

One other commonly used game is called the tragedy of the common. This game is played by several players that share the same resources. In this game it has been shown that if the players act independently in their own best interest they would eventually deplete the resource, which is not in their long term best interest. Elinor Ostrum, a political scientist, has researched the matter looking at various communities around the world that share resources. She actually found that it’s true and that most communities have depleted their resources, which was not in their best interest. But she also found something else; there were many communities that didn’t deplete the common resources and managed to sustain a long-term preservation of these resources. By looking closely at communities that succeeded vs. communities that failed she discovered that communities that succeeded in preserving these resources managed to establish institutions for collective actions, and that collaboration was the main reason for the positive outcome.

Robert (Yisrael) Aumann and Thomas Schelling received the Nobel Prize in 2005 for their work on collaboration in games (“The tile of Aumann’s work is “Conflict and cooperation through the lens of game theory”) and addressed the question of which games would promote collaboration and why people chose to collaborate. Part of Aumann’s findings was that multi-player repetitive or infinite games promote cooperation between players because people learn to collaborate in time.

Another contributing factor to cooperation is introduced by some irrationality in the players, meaning that perfection and uniform thinking does not lead to collaboration. There is a need for some of the players to act differently than others given the same situation in order to promote collaboration or, as George S Patton described it, “If everyone is thinking alike, then somebody isn't thinking.” -- George S. Patton.

So if we look at all of these examples we can see that in multi player infinite games with some irrational behavior (like trading for example) sharing and collaboration are always in our best interest. In my mind we still haven’t begun to realize the true potential of collaboration and the payoffs it would yield but we can clearly identify that collaboration, in the eyes of game theory, leads to greater success.

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Be sure to read the full risk disclosure before trading Forex.  Please note that Forex trading involves significant risk of loss. It is not suitable for all investors and you should make sure you understand the risks involved before trading. Performance, strategies and charts shown are not necessarily predictive of any particular result. And, as always, past performance is no indication of future results.

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Be sure to read the full risk disclosure before trading Forex. Please note that Forex trading involves significant risk of loss. It is not suitable for all investors and you should make sure you understand the risks involved before trading. Performance, strategies and charts shown are not necessarily predictive of any particular result. And, as always, past performance is no indication of future results. Investor returns may vary from Trade Leader returns based on slippage, fees, broker spreads, volatility or other market conditions.

Probably one of the earliest applications of psychology to the financial world was the Prospect Theory in 1979.  Two psychology professors, Amos Taversky and Daniel Kahnman, proved that people when faced with a financial decision will not necessarily make the most logical selection. This is because of the way we think about our prospects and not the big picture scenario when making financial choices. Kahnman was awarded the 2002 noble prize in economic for his work in the field.

What they proved is that people are affected by losses more than they are affected by wins when they are required to make a financial decision. For example, when a person is asked to choose between receiving $100 or tossing a coin to receive $210 most people would prefer to take the $100 even though the most cost effective option would be to toss the coin (This is because the probability of winning is 0.5 so the utility function for tossing the coin is 0.5*$210 = $105 which is greater than the other option of $100). Another interesting thing they have found is that that the opposite behavior happens when they are faced with losing money. Most people would prefer to toss a coin with the 50/50 chance of losing $210 as opposed to just giving away $100, which again is not the logical option.

So to make a long story short – people like to make as much money as possible without risk and take more risk than necessary when they’re about to lose money.

We can see one application of this in real life trading where most traders tend to close winning positions sooner, take their profit and avoid additional risk. On the other hand they could leave loosing positions longer, take additional risk for the chance that the market would change direction (which rarely happens) and they wouldn’t loose as much.

There are many other applications of this behavior in real life trading. Stay tuned for the next post in this series, coming soon.

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Be sure to read the full risk disclosure before trading Forex.  Please note that Forex trading involves significant risk of loss. It is not suitable for all investors and you should make sure you understand the risks involved before trading. Performance, strategies and charts shown are not necessarily predictive of any particular result. And, as always, past performance is no indication of future results.

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Be sure to read the full risk disclosure before trading Forex. Please note that Forex trading involves significant risk of loss. It is not suitable for all investors and you should make sure you understand the risks involved before trading. Performance, strategies and charts shown are not necessarily predictive of any particular result. And, as always, past performance is no indication of future results. Investor returns may vary from Trade Leader returns based on slippage, fees, broker spreads, volatility or other market conditions.

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In the following few posts I’ll try to give a few examples of how game theory is applied to trading the forex market hoping to shed some light on the mathematical aspect of trading and the value of collaboration in this market.

The most common statement I hear from traders is “The forex market is an efficient zero-sum game market and therefore sharing information may expose my edge.”

So what is a zero-sum game and how do I know if the above statement is true? Let’s start by talking about game theory and the different types of mathematical games.

Game theory is relatively young and quite complex branch of mathematics applying math to structurally define strategic situations in games. Mathematical games are most commonly defined by the number of players, duration of the game, game sum (more on this in a minute), simultaneous/sequential, and information flow. So for example chess is a two player, finite, zero-sum, sequential perfect information game because there are two players, the game ends when one of the players wins, only one player can win, the players take turns in playing and each player sees the actions of the other trader. (This is what defines perfect information flow.)

Economy, in general, is a multi player, infinite, positive-sum (Hold on, I promise I will explain why this is), simultaneous, imperfect information game.

Zero-sum games are games where a win by one player means a loss by the other player; the term has little to no relevance when describing multiplayer infinite games. The reason economy is a positive-sum game is because it’s an infinite game where value is built throughout time due to value growth in the overall economy.

So what about retail forex trading? Is forex a zero-sum game?

The short answer is that it doesn’t really matter because it’s an infinite, multiplayer game. The long answer is that it’s not a zero-sum game and it largely depends on the way the broker operates. Brokers that operate as a dealing desk, for example, (also known as market makers) mathematically resemble the way a bookie operates in a horse race. This becomes a negative-sum game because mathematically the players in the game are all traders using the same broker.

So what is the best strategy to play in a multiplayer infinite game with imperfect information – like the retail forex market? – More on that in future posts …

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Be sure to read the full risk disclosure before trading Forex.  Please note that Forex trading involves significant risk of loss. It is not suitable for all investors and you should make sure you understand the risks involved before trading. Performance, strategies and charts shown are not necessarily predictive of any particular result. And, as always, past performance is no indication of future results.

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Be sure to read the full risk disclosure before trading Forex. Please note that Forex trading involves significant risk of loss. It is not suitable for all investors and you should make sure you understand the risks involved before trading. Performance, strategies and charts shown are not necessarily predictive of any particular result. And, as always, past performance is no indication of future results. Investor returns may vary from Trade Leader returns based on slippage, fees, broker spreads, volatility or other market conditions.