It has been a while since I wrote an article. I therefore owe you, my readers, a sincere apology. With the ever looming lockdown coming for Kuwait in February 2021, I decided to take a deep breath from university studying — to pause, reflect, and continue writing once again.
An interesting subject I’ve recently came across was ‘Game Theory’, and therefore I’ve selected it as the first article to once again revive the streak of article writing in 2021. As you probably already know, every topic I come across, I tend to find a way to link it to the financial industry — as mathematical finance seems as of yet, to be my calling.
Let’s begin —
What is Game Theory?
The word game appears with a few definitions in a limited set of contexts. The one we are immediately interested in is the one directly associated with it.
A game in game theory is any interaction between multiple people in which each players payoff is affected by the decisions of others.
The key word here is decisions. At the end of the day every move is governed ultimately by a taken decision. And a game is nothing but a decision making situation broken down further into smaller decision making situations.
There are three distinct decision making situations:
- Deterministic situation: it is a decision under certainty with full information
- Probabilistic situation: it is a decision under risk with partial information which is described by a probability distribution
- Uncertain situation: it is a decision under uncertainty with incomplete information
Now these interactions, these games, come in two main categories,
- cooperative games: everyone benefits by working together. eg. driving on the right side of the road.
- noncooperative games: eg. the prisoners dilemma
Of course not all games are limited to the aforementioned categories. An example of such games are zero-sum games, where one players win is another’s loss and vice versa. However, games that are non zero-sum allow for mutual gains and losses.
There exists two important properties of any game:
- The Dominant strategy: strategies that are better than other strategies for one player, no matter how the opponent may play. They are not optimal if you are in a game where each player has a dominant strategy. so in Prisoner dilemma dominant strategy is betray & stay silent.
- The Nash equilibrium: is a state that no one person can improve. Every finite game has at least one Nash equilibrium.
The purpose of utilizing the framework of game theory would be to simplify real world interactions and to boil them down into models whereby we can predict the moves/actions available and positions of each player to secure the best outcome.
Before we even begin building said models, it is imperative to understand the 5 key conditions (which at times double as assumptions) that must be fulfilled prior to creating these models.
The 5 conditions:
- A game needs to include multiple player (2 or more)
- Each player has two or more choices or sequence of choices
- All possible combinations of these decisions are plays resulting in a clear outcome = win or lose
- Its clear how you can win or lose. participants will gain or lose something depending on the outcome.
- The players know the rules of the game as well as the payoffs of other players.
- The players are (perfectly) rational and sensible people acting out of their own self interest.
Principal agent problem is when one person is allowed to make decisions on behalf of another person. The first person is likely to prioritize their own interest to reach their own goal.
Before we go any further It would be wise to familiarize ourselves with the terminology used in game theory:
- Players: eg. player A & player B or company 1 & company 2
- Strategy: a course of action taken by a player. There are two types of strategies: a pure strategy — one particular strategy, and a mixed strategy.
The denoting of games sheds light on the nature of the game too. For example, sequential games (where later players have some knowledge of earlier actions) are denoted by decision trees. Whilst simultaneous games where both players move or do not move together are denoted by payoff matrices (eg. shown below).
Instead of delving deeper we will jump straight to how we can tie game theory to price fluctuations.
The Keynesian Beauty Contest
In the 1930’s there was a popular newspaper contest that required people to guess the most attractive face out of the 100 photographs. The readers were expected to select 6 out the 100 which they personally thought were the prettiest.
Later the newspaper would collect all submissions, count the results, then they would see if your submission contained the popular choice. If it did contain the popular choice, then you were automatically placed into a draw to win a prize.
Strategies to approach this game (process of reasoning):
- The Naïve Strategy: choose the faces you personally find attractive
- Base-level: Assume every player chooses the faces they find the prettiest, so you select the faces you think everybody else finds more attractive
- First level: Assume every player chooses the faces that they think everybody else will find the prettiest
- ….
- ..
- .
- Infinite-level: The Nash Equilibrium (not realistic as humans have bounded rationality not perfect rationality)
etc.
Keynes saw that the stock market worked in a similar way to that of the beauty contest; where everyone wants to buy before everyone else, and wants to sell before everyone else. Essentially the similarity between Keynes Beauty Contest and the stock market seems apparent in short term trading, where the ‘edge’ is the previously mentioned strategy of assuming what everyone assumes the most attractive stock is to a high/close to infinite level rather than focusing on the fundamentals of the stocks.
Of course the Nash equilibrium strategy is not entirely possible in reality, and thus we ought to run experiments rather than simply reasoning using game theory.
This concept is rather discrete. It does not account for changes. Actually what we ought to ask, more accurately, is how do we account for the changes in strategies over time?
Evolutionary Game Theory
Building off the fact that humans possess bounded rationality, we must disregard models that assume perfect rationality. This means we have succumbed to scrapping off all simplistic models. Doing without the perfect rationality assumption means we allow players to make mistakes and thus provide a more realistic model of reality.
Another alteration is to begin to apply evolutionary game theory which studies the players who change their strategies over time based on behavioral rules that are usually not rational. To model such evolutionary games we use Markov Chains where the state variable is the strategy being used or how the game was being played in the past (historical data).
It is appropriate to link an article of mine where I talked about Markov chains in stock market trends.
One should therefore identify the key players in any trade and then their respective payoff function for each action available to them per their goals and motivations.
An area of focus in evolutionary game theory is replicator dynamics where a replicator creates more or less accurate copies of itself. In game theory the replicator can be a strategy in a game. In mathematics, the replicator equation possesses the following properties:
- deterministic
- non-linear
- non-innovative game dynamic
It is used to model replication in the form of a continuous differential equation.
where,
- ẋᵢ; growth rate of proportion of the population playing strategy i
- xᵢ; is the proportion of type i in the population, which the current frequency of strategy
- [fᵢ(x)-Ø(x)]; own fitness relative to the average
- x=(x₁, x₂,…, xn); is the vector of distribution of types in the population
- fᵢ(x); is the fitness type i which is dependent on the population, where fitness is interpreted as rate of reproduction
- Øᵢ(x); is the weighted average of the fitness of the n types in the population, which is the average fitness
Since populations are generally finite we can utilize the discrete version. We can further simplify by assuming fitness to be linearly dependent on the population distribution, hence forth re-writing the equation as follows:
where,
- the payoff function A: holds all information for the population
What we can use the replicator equation to:
- Determine the expected payoff from the payoff function A.
- Identity the existence of steady states (asymptotically stable).
- Show us that in the end if it converges it will converge to the Nash equilibrium.
There are papers such as this one by Benoit S. Montin (2004) about using “the replicator equation in a continuous time setting and for homogeneous and finite population playing a symmetric game”.
Conclusion
As of yet I am still trying to get to grips with the underlying mathematics behind game theory and more precisely, evolutionary game theory and replication dynamics.
However what I can conclude from this short but eye-opening (and far from over) intellectual adventure is that there are many different ways to attack the problem of having uncertainty and how to deal with it.
No matter which way we go, we always carry a multitude of assumptions that can ruin our models. At the end of the day whether we target the problem of prediction and thence decision making from a stochastic or game theoretic manner, we end up using similar tools to deduce very interesting answers.