## To what extent does game theory help the society to explain financial markets

Game theory is a mathematical approach by which two individuals make decisions about a situation which can have several outcomes, based on their interaction with each other. The objective of both players is to maximise their returns, thus cost and profit made out the situation can only be properly calculated based on the decisions made by the individuals. The description of the game is best understood by representation of the game, in which each player simultaneously chooses a strategy, and combination of strategies chosen by the players determine a payoff for each player which can be understood through a pay off diagram.

John von Neumann and Oskar Morgenstern introduced game theory in 1943 with "Theory of Games and Economic Behaviour." The major objective of Game Theory was to find mathematical answers to economic problems.

As per economic theory, producers often take supply and demand as the major ingredient to increase profit. But it ignores the strategies taken by other producers in order to improve their market share. Game Theory helps in anticipating the various strategies from different players and take into account all those strategic interactions.

1.1 Example on Game Theory:

The above mentioned paragraphs can be simplified through an example.

A soldier who has been posted around the border is going to take part in a battle. There can be two possible outcomes of the battle (victory or defeat). Similarly there are two possible outcome from the battle for the soldier (die or survive). So the soldier can either fight or run away, where the later has the highest probability of seeing the soldier survive. But the fact is that every other soldier is also thinking along the same line, therefore in that case if the soldier decides to cooperate and fight then he will surely die. Now in the mean time the army general has come up with order which states that anybody found fleeing from the battle field will be considered a traitor and will be shot dead. So the best possible chance for the soldier to survive in the battle is to fight and win.

This example states how the soldier had come up with his strategy to survive from the battle field and how that strategy of the soldier is countered by the army general.

Game Theory assumes that all individuals are rational and decisions taken by them will be rational. So what may seem irrational to the rest of the world is perfectly rational from the point of view of Game Theory.

We are going to take another practical example where Game Theory is applied in Business.

There is a hypothetical company A which is a market leader in its own area. The market is monopolistic in nature and so it is A which calls all the shorts. Company A makes handsome profit of 300,000 pounds per year. There is a new company called B which is trying to enter in this market.

Now let us take this scenario forward.

If B enters the market, A has two options to tackle this issue

a) Either A will allow the new entrant to enter the market and compete or

b) Forcefully not allow B to enter through price war.

Now a separate firm has come up with market analysis that if B enters the market then both firms A and B will make a profit of 100,000 pounds.

In case of price war, company A will make a net loss of 100,000 pounds and company B will make a net loss of 200,000 pounds.

So this prompts to form a game tree.

Profit of A = 100,000 pounds

Profit of B = 100,000 pounds

Accepted by A

Enter market

Loss of A= -100,000 pounds

Price war Loss of B = -200,000 pounds

Company B

Enter market Profit of A = 300,000 pounds

Profit of B = 0

So the best case scenario for both companies is to exist mutually and peacefully and do business. Though we have used a very simple example but a real life scenario might have several stages of decision making and more factors would get involved. However the analytical framework will remain same even in that case.

In the example above we see an optimal strategy is not only going to benefit a particular player but will be of help to all the players involved in the game and that is why we call it an Optimal Strategy and is a solution to the game.

Let us take another example where we will be using game table to analyze the results of Game Theory. This is a very famous example called the Prisoner’s Dilemma which will be solved form the light of Game Theory.

There are 2 criminals caught by police for interrogation for their alleged involvement in a crime. The police give them the following pointers.

a) If both confess they will be sentenced for a period of 10 years in jail.

b) If only one of the criminal confesses, he will get 1 year sentence but the other criminal who has not confessed gets a 25 year term in prison.

c) If neither of them confesses, then both get a 3 year term in prison.

Taking the pointers forward a decision tree can be formed for this example above.

First prisoners decision

Confess Not confess

Second prisoners

decision

Confess

10 years

25 years

Not confess

1 year

3 years

Both the prisoners may have dominant strategies which will be the most benefitting to each individual.

Or they can come up with a strategy where both the players can reap out benefit and exist mutually.

1.2Terms associated with Game Theory

Pure strategy: It can be defined as a strategy which comprises of various authentic decisions that can be taken for various cases devised while framing the strategy.

Mixed Strategy: Devises a probability to each pure strategy which helps the player to comprehend and make a choice on a random basis.

Nash Equilibrium: Two players involved in a game have a proper know how of the equilibrium strategy of each other, and none of them are able to maximize their pay off. In that case the strategy taken by the players is adopted and this is of help to the all players involved. This is called Nash Equilibrium.

1.3 Application of Game Theory to analyze Financial markets

Introduction

After the introduction of Game Theory by John von Neumann and Morgenstern in 1944, the theory found widespread application not only in economics, but also in various other branches of study like computer science, engineering, biology, political science, international relations and even philosophy.

As the global financial markets became more and more unified with increased number of market participants and larger intervention from the policy makers and newer sophisticated trading methodologies, Game Theory became very popular even in the financial markets.

The application of Game Theory in financial markets found wide acceptance even from the stalwarts. The Economist (1996) came up with these lines “managers have much to learn from game theory — provided they use it to clarify their thinking”.

The Wall Street Journal (1995) which is considered the most revered of all in the world of financial markets said “game theory is hot”.

This clearly tells how Game Theory has been revolutionary in financial markets not only in simplifying various financial theories but also in the latest algorithmic trading methodologies which are applied by the leading Hedge Funds and large financial institutions, for financial asset pricing and price discovery.

There is a very famous line from the great scientist Sir Isaac Newton “I can calculate the motions of heavenly bodies, but not the madness of people” after he lost £20,000 in the South Sea Bubble in 1720.

1.4 Application of Game Theory to Model Market Uncertainty

In the insane world of financial markets, price of assets fluctuate based on the mindset of various market participants. The probability model is considered the most popular in figuring out uncertainties associated with any event. In case of stocks and financial assets, price tends to move up and down owing to various fundamental impacts and preferences of market investors. So the fluctuation of asset prices can be considered as a random event taking the probability theory forward. (Smith, 1964)

When Game Theory is applied, the price fluctuation in financial markets is not considered a random event. Since the uncertainty in prices are brought about by people who are considered rational entities so it is assumed that the price changes are various choices made by rational entities and not as unpredictable events.

There is another very popular theory associated with financial markets which is known as the Random Walk Theory that explains asset price fluctuation is a function of a statistical distribution model. (Avery and Zemsky 1996)

1.5 Approach of an investor from the standpoint of Game Theory and Probabilistic model

An investor with a probability theory approach will invest, looking into historical data with a view that that the future is a function of the past and therefore will resemble the past. Similarly the investor who applies Game Theory is looking at the position of other investors and traders in the stock, and makes prediction based on reasons rather than looking at statistical data only.

The view of the investor with Game Theory can be analyzed with an example.

When a stock has fallen a lot, the normal notion is that the informed investors (investors who are smart and have insider news about the stock and have the ability to influence prices) have shunned the stock and moved out. As a result the stock finds seller at every level and it is only the retail investors (considered weak hand) who come in and buy. As the sellers are substantially higher than the buyers the short build up (selling the stock in the derivative market) in the stock is more pronounced. Now when the stock moves up it will result in a short squeeze (where the short sellers will have to buy the stock in order to avoid losses). As a result the stock will move up. The investor who applies Game Theory will look to buy the stock as he will feel the market participants have gone wrong and so the stock will continue to move up till there is a complete short squeeze. (Vila, 1989)

1.6 Application of various pricing model for analyzing asset prices from the perspective of Game Theory:

As the financial markets have gone through an evolution over the years, various models have been devised to determine future prices of financial assets. Some of the most popular model used for valuing future prices of asset are

1) Fundamental analysis

2) Technical analysis

3) CAPM (Capital Asset Pricing Model)

The financial market comprises of various players and each player has his own strategy of using the historical data to figure out the expected future value of the asset. When Game Theory is applied the strategy of each player distorts the information available. This theory of distorted information was brought forward by the authors Wiszniewska-Matyszkiel which applies Game Theory.

As per this theory there are large numbers of players in a mature stock exchange where none of the players have the ability to bring about any effect in the price of an asset.

The various pricing techniques are discussed in short.

Fundamental Analysis: As per this approach the stock prices are the discounted value of expected future dividends. As per Fundamental Analysis the asset prices are expected to follow the fundamental value and any deviation of price brought by market volatility is a temporary phenomenon as asset prices tend to move back to their fair value. (Miller and Modigliani 1961)

Application of Game Theory in Fundamental Analysis:

As per the model proposed by Ross (1977) a high growth firm with high debt will not be dishing out dividend like a firm with steady growth and similar level of debt. This high retention of funds by a high debt high growth firm will help it in near future. As per Meyers (1984) if a manager of a firm has a better idea regarding the prospects of a firm available to any other sources, will definitely look at issuing fresh capital for investments through retained earnings and debt rather than equity. The firm will not look at issuing new equity in order to raise capital especially when equities are undervalued. (Bhattacharya, 1979) Rather they would prefer to issue fresh equities when equities are fair to expensive in terms of their valuation. Thus this entire decision making process in helped from the stand point of Game Theory as information available in markets are asymmetric in nature.

Technical Analysis: Is basically analyzing the future price movements of asset classes from past historical prices available. The basic assumption of technical analysis states prices move up in trend and prices tend to reflect the future of a company.

Game Theory can be applied to fine tune the results of Technical Analysis.

The model used in Technical analysis uses probabilistic approach where the prices tend to be random values out of a statistical distribution. Using Game Theory a trader can take various parameters into consideration and form a risk reward ratio matrix from the possible outcomes of his decisions. The one with the highest chance of generating profit is taken into consideration.

CAPM: This model explains the relationship between systematic risk and expected return which is used in measuring the price of the security. CAPM can be defined as the return expected out of a portfolio or a stock is the risk free rate which is prevailing in the market and risk premium. (Sharpe and Lintner,1965)

Re= Rf + b( Rm-Rf)

where ,

Re is expected rate of return

Rf is the Risk Free Rate

Rm is return generated from market over a period of time and

b is the systematic risk.

However the CAPM model has a few flaws associated with it.

1) The treatment of CAPM is more ex ante. As most of the parameters used are theoretical in nature the back testing of the model in reality will lead to error. (Ferson 1995)

2) Calculating Beta value for various companies and project is a very difficult task and may often lead to wrong findings as beta which happens to be the systematic risk tends to change in various economic conditions.

3) Another assumption of CAPM that may be taken as faulty states that all investors tend to have similar investment horizon which is never the case. An investor may have an investment time frame of 1 year and the other may have a time frame of 2 years. However as per theory both of the investors have long term view in terms of their investment.

4) The risk free rate denoted by Rf is taken to be constant. But in case of change in economic condition the risk free rate can often see changes and may remain volatile.

To eliminate these flaws Game Theory has been used

The changes put forward in CAPM from the light of Game Theory states that:

1) The Game Theory eliminates all assumptions on preferences and likes and gives utmost emphasis on efficient market hypothesis.

2) The formula for CAPM which uses an exact equation between theoretical quantities is not taken into account while applying Game Theory. Rather an approximation is taken on account of empirical values.

3) The expected theoretical values like variance, covariance are replaced by certain empirical values.

The Game Theory model turns the CAPM model into ex post rather than ex ante. So using the Game Theory in CAPM, the asset prices cannot be obtained in advance.

The classical CAPM model was difficult to test as the values used were mostly theoretical. With the implementation of Game Theory the empirical values so used to back test as the theory implements efficient market hypothesis.

Extending CAPM from the light of Game Theory:

A financial market is chosen whose with an index Is. The return from the index is considered as Ris.

The return from holding a security or financial asset S is Rs.

Zm^2 is practically the variance of the index return Ris tested out from the data obtained over a period of time.

and Zsm is the covariance of the return of index is and return of S.

Now taking these tested practical values we construct the new CAPM equation :

Rs~ Ris - Zm^2 + Zsm

The calculation is based on N days

Rs= 1/N ∑Sn (here the summation is between n=1 to n=N)

Ris= 1/N ∑Isn (here the summation is between n=1 to n=N)

Zm^2= 1/N ∑Isn^2 (here the summation is between n=1 to n=N)

Zsm = 1/N ∑Isn*Sn (here the summation is between n=1 to n=N)

This approach of CAPM is ex post and is more useful in application to real life scenarios. (Vovk et al, 2002)

1.7 The application of Game Theory in order to prevent stock market manipulation

With the growth of financial markets stock markets are no longer a place to trade securities but are a serious source of capital for a nation.

To safeguard the interest of the investors regulatory authorities act as 'watchdogs' to prevent any malpractices in the financial market. After the recent global mayhem of 2009, market regulators have taken various steps to bring investors back to financial market. (Hart et al, 1986)

To improve the overall functioning of financial markets the market regulators have taken various steps to woo back investors like:

a) The transaction costs have been reduced for investors.

b) The long term capital gain taxes have been reduced.

c) Eliminating various malpractices associated with security market and encourage long term capital formation

All these steps taken by the regulators have once again increased the trading volumes back in the exchanges. (Jarrow, 1992)

Market manipulation is a major issue associated with financial markets

Manipulations are done by a group of people who have access to various key information by which the prices of assets are influenced.

A large block deal involving buying and selling of large number of securities of a company will definitely result in a change in price of that security. So the informed group like promoters, large corporate houses, and institutional investors can strongly affect the prices of securities by the various transactions. The illiquid securities (thinly traded shares with very low volume) are the ones which tend to get most affected by this. (Bommel,2003)

In order to reduce such malpractices the market regulators have recently taken various steps to eliminate manipulations in stock markets.

a) The Indian Market regulator SEBI has made it mandatory for companies to declare the details of the share holders to the exchanges.

b) The companies have to inform the exchanges regarding any substantial big transaction deals called block deals.

c) The promoters of a company are not allowed to trade in securities till the price sensitive news associated with a company is out in public.

d) Many volatile stocks which have been subjected to bear cartel (where certain informed operators created huge shorts in the derivative segment and in this process take the prices down) have been moved out of the derivative segment.

e) Introducing circuit filters in securities to keep the volatility under check.

f) In 2011, the German market regulators had banned short selling of securities to minimize any chance of manipulation in the DAX (German stock exchange). However it was not taken with good spirits by the market participants and the financial markets across the globe sold off.

Now let us take a game theory approach as to how a manipulation can be stopped in a financial market by the vigilant actions taken by the market regulator. (Allen et al, 1992)

There are 2 rational entities:

One of the rational players is an investor who looks to manipulate the financial market by using various price sensitive insider news and information.

The other player is the market regulator.

The assumption made over here is all players are rational and each player know that the other player is rational.

Let us draw the decision tree:

Investigates the matter

Manipulates

Does not investigate

Does not manipulate

Profit made by manipulator

1) On manipulation he makes higher profit P+M ( where M > 0)

2) Without manipulation makes profit P

Profit made by the market regulator

1) If investigates the matter the regulator imposes a fine of all the profit on the manipulator so the regulator makes P+M. In the process of investigation the regulator has to bear certain cost for investigation.

Let us take the investigation cost as I

2) In case of no investigation the regulator loses on this opportunity to make money and makes no profit.

This is a pay off diagram for the various scenarios in the problem

Manipulator

Manipulates No manipulations

Market Regulator

Investigates

0

P+M-I

P

-I

No investigation

P+M

0

P

0

So from here we get an idea that if the market regulator is on its toes it can not only remove all malpractices from the financial market but can also make money in the form of fines imposed on the manipulator.

1.8 Exploiting Game Theory for Profit in the Stock Market

Till now from the light of Game Theory it has been seen that the decision taken by a player not only depends on his own choice but also on the decisions taken by other person who are involved in the game. Now taking this idea forward it will be seen how a smart trader makes money out of the financial market through trading.

But prior to that the idea of Risk arbitrage has a significant weight in analyzing how profit can be made out. So Risk Arbitrage is a relatively low risk strategy of turning a profit by carefully analyzing the market situation through the lens of a constantly changing environment in which one must predict the actions of other players in order to figure out the best strategy for himself. One way to do so is to analyze historical data of past trades to figure out the trend. According to Prof Brown while entering into a trade a trader must identify the company he wants to take position in. Then he should go forward and analyze the goal for which he has entered the trade. The next step is finding out the pay off diagram for each player involved in the trade. However the author feels that a mere use of Game Theory will not help in making profit. Hence a smart trader should adopt Game Theory as well as Statistical analysis while taking his trade forward. But since the various factors at play in the stock market have a very big bearing on the global economy so one should have a clear idea about the overall macroeconomic and micro economic factors. (Brown, N.D.)

The Game Theory approach to make profit form a trade can be taken forward with the help of another example.

There is a hypothetical case where a company X is going to acquire a company called Y by the share swap mechanism. The overall mechanism involves swapping one share of X with two shares of Y. The price of X's shares is 50 pounds and that Y's shares are 20 pounds. So in order to take this merger forward the fair price of Y shares should be 25 pounds. However the share price of Y is currently quoting at 24 pounds. So a riskless arbitrage opportunity appears where a trader can buy 2 shares of Y and short 1 share of X and in the process make a riskless profit of 2 pounds. So by applying risk less arbitrage model a profit of 2 pound is made. However the process is not always riskless as there are chances that the process of merger might not work out which may result in a loss for the trader.

From the light of Game Theory we need to find out the decision makers involved in the total process of merger. So the decision makers involved could be the management of X, management of Y, investors and market regulators. Thus each decision maker will have his own pay off diagram. However the involvement of so many decision makers makes the situation complex. To make it simple Game Theory is applied with Statistical Analysis.

In 1991 Time Warner the world’s largest media and Entertainment Company known for its brands like Fortune, Time, People and Sports Illustrated had decided to raise fresh capital through a right issue to existing shareholders.

The term of the offer was made in such a way that ultimately forces the existing shareholder.

It was decided that

a) If 100% shareholder participation takes place in the right issue then the price of right would be fixed at $105 and the number of shares for each participant would increase by 60 % after the offer closes.

b) If 80% shareholder participation takes place in the right issue then the price of right would be fixed at $84

c) If 60% shareholder participation takes place in the right issue then the price of right would be fixed at $64.

However the decision was not taken positively by the shareholders and the stock tanked over 25% in a few days after the announcement. The stock fell from $105 and finally settled at $85 in the NYSE (New York Stock Exchange).

Let us construct a pay off matrix where we have taken the price purchase of the share at $90 and decision to take part in the right issue by 100%, 80% and 60% participants from the three scenarios decided upon by the company.

Cost pay off matrix for the problem

Decision alternatives

Participation from Share holders

60% participation

80% participation

100% participation

Market price

$90

$90

$90

Right offer participation

$63

$84

$105

So clearly it can be seen that if the market price of the stock price can be brought down below the prices of right then the shareholders can be forced to participate in the deal. Instead the shareholders can purchase shares of the company and increase their stake in the company directly from the market and therefore not take part in the right issue as buying form the market offers a much cheaper alternative for the shareholders. Thus the fall of share price was logical enough to counter the decision of the firm.

Taking the best possible scenario for the shareholders the price of the right issue should have been fixed at $63. However from the valuation perspective the price around $80 was cheap and fair for the deal to go forward.

This move completely took Time Warner off guard and the company decided to fix the price of right issue at a fixed price of $80.The market gave a big thumps up to the decision and the stock soared up in the NYSE (New York Stock Exchange) after the news broke out.

So this clearly gives us an idea as to how Game Theory was used to analyze the various outcomes of the entire event and how the best outcome was obtained from the perspective of both parties involved.

1.9 Major Disadvantages of Game Theory

Like any other model Game Theory has some imitations that make it less practical to use in some cases.

a) It has been found that it is wrong to assume that a player has a proper know how about their own strategic pay off and the pay off other players involved in the game.

b) It is very difficult to analyze a game that involves large number of players where a mixed strategy is employed. The pay off diagrams becomes very complex due to large number of alternative decisions that can be taken.

c) It is impossible to find out solution for all problems from the stand point of game Theory.

d) Though Game Theory says each player has a proper know how about the decision of the other player but practically the other player may have various alternatives at his disposal which the others may not no as there is written down rule as to which strategy will be employed by the other player.

So though Game Theory has some flaws associated with it and might not give practically feasible solution to a problem but its ability to find out the best case scenario for a problem cannot be ignored.

2.0 Conclusion

Game Theory has had a huge impact in the decision making process of any event. Game Theory became so popular that the life of famous Game Theorist John Nash was turned into a movie “Beautiful Mind” where the lead role was played by actor Russel Crowe.

The beauty of Game Theory is that it tries to assess the world from this very principle that “If we were all better people the world would be a better place." This was the basis of solving the famous problem of Prisoners Dilemma. (Hwang et al, N.D.)

Till the evolution of Game Theory, there was practically no alternative by which one could have figured out and assessed the thought process of others involved. This practise not only helps in keeping one much ahead in a competitive environment and come up with the best possible result from any outcome but can also see one grow morally. It helps a person grow morally and makes one a better human being as the person can understand others much better. If an individual analyzes others from the perspective of the other person rather than form the perspective of own self then much of the problems across the globe would not exist and the world would be a better place to live. People would understand each other in a much better way and that would definitely eliminate hatred, treason and treachery.

Thus it can be clearly seen that Game Theory will continue to remain as an added tool for analysis of any problem as it opens up a whole new facade of logical thinking and decision making.

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