Mathematics

The science of interaction and decision-making: Game Theory

From the very beginning, we have tried to predict the future. Prediction is a messy business, and astrologers are seldom genuine. However, scientifically speaking, prediction is absolutely essential. From weather to strategic interactions, prediction is present everywhere. It is often impossible to predict complex, real-life phenomena using simple mathematical equations. We need a new concept for that.

On one hand, we have a new science – chaos theory – which studies complex, unpredictable phenomena in terms of strange patterns called fractals and some universal laws regarding them [1]. Chaos theory studies systems which are unpredictable, and slight changes in the initial conditions of which can lead to drastically different outcomes. With its wide-ranging applications, chaos theory is indeed a very promising science.

On the other hand, another science – game theory – deals with the mathematics of interaction between players in a game. It is essentially a science of decision-making. Life is indeed analogous to a game, and thus, game theory also has wide-ranging applications in economics, biology and beyond. Game theory was first developed by John von Neumann and Oskar Morgenstern, and further developed by many mathematicians like John Nash and Lloyd Shapley [2].

It might seem appealing to think of decision-making in terms of neurology and psychology. While those fields can offer us great insights into human decision-making, we must turn to mathematics if we are to gain a rigorous understanding of decision-making at a fundamental level. An interesting point to note here is that mathematics cannot predict the whims of the human mind, and thus mathematics alone cannot explain human decision-making to a great extent. To illustrate this, mathematics makes no difference between the following two scenarios.

In the first scenario, we have an out-of-control trolley running down the rail tracks at a tremendous velocity. In front of the trolley there stands four railway workers who would die if the trolley is not diverted to a different track. There is only one other track, and you stand very near to the lever. Thus, it is an easy matter for you to switch the lever and divert the trolley into the other track. However, there stands one railway worker on this second track, who would die if you divert the trolley. What would you do, save four people and kill one, or let things proceed on their own?

Figure 1: The first scenario of the trolley problem.

In the second scenario, everything is the same, except that this time there is only one track, some way down which four helpless railway workers stand. You are on the top of a tower, located adjacent to the railway track. In front of you, on the top of the tower, stands a fat man. If you were to push the man from the top of the tower, he would fall on the track, before the trolley. His body would suffice to stop the over-speeding trolley, and save the workers. However, he dies. Would you push him?

Figure 2: The second scenario of the trolley problem.

A logical reasoner should view both the situations as the same. In reality, as researchers demonstrate, ethical questions come into consideration in the second case. The first case is a maths problem. One death is better than four. So is the case with the second scenario. However, here you are physically interacting with the person. That is direct murder. This thought experiment is known as the trolley dilemma, and though most people when introduced to this dilemma answer in the affirmative in the first case, almost all of them hesitate or answer in the negative in the second case [3]. This is because when you are thinking of physically pushing a person to his death, brain regions involved in producing emotions and a sense of conscience get involved. It is no longer a maths problem. However, to a logician, it should still be. This example was to illustrate that there are certain subtle aspects which are, as of yet, inaccessible to game theory. Game theory, for instance, demands that players act to maximize their profits. Players, thus, should take risks only when the expected return is high, and the risks/losses are affordable. However, this does not explain why lotteries are so popular, despite the fact that the expected value of a lottery ticket is almost always lesser than the ticket price. Another thing to realize is that the value of something cannot be quantified universally. To a person who already has a million dollars, another million dollars is of little value. Even half that amount is more valuable to someone who has very little money. Once your basic needs are satisfied, the value of further money drops non-linearly. However, in general, game theory offers us a great understanding of strategic interactions and corresponding decision-making.

Competition has been scientifically studied since Darwin’s time. Be it the survival-of-the-fittest concept in evolution and natural selection, or competition to gain market share in a modern industry, competition is omnipresent. “The theory of games is a mathematical discipline designed to treat rigorously the question of optimal behavior of participants in strategic games and to determine the resulting equilibria” [4]. Jargon? Essentially, all it means is that game theory is a science to study and predict interactions between players in a game. Not only that, there is much at stake in these so-called “games”, which are not something you casually play with your friend over tea. In such games, each participant is striving for the greatest advantage in situations where the outcome depends not only on their actions alone, nor solely on those of nature, but also on those of the other participants whose interests are sometimes opposed, sometimes parallel to their own. A game, more precisely, in this context, can be defined as a set of circumstances that has a result dependent on the actions of two or more decision makers (players). A move in this game is a point where the players are faced with choices. In such games we say an equilibrium is reached when both (or more) players have made their decisions and an outcome is reached [4].

Game theory deals with cooperative and non-cooperative games. In cooperative games, every player agrees to work towards a common goal. In this case, the gain of the entire group (coalition) is more important than personal gain. The question addressed by game theory is how much each player should contribute to the coalition, and how much they are to benefit from it. The contribution (more specifically, the marginal contribution) of each player is determined by what is lost by removing them from the game. If everything remains the same after the removal of a player, it means that the player contributes nothing to the game, and should in turn receive nothing (although there are exceptions to this). In non-cooperative games, in contrast, opponents play independently to maximise their personal gain [7].

There are some assumptions in game theory [5]. For instance, the number of participants in the game must be finite. All participants are assumed to act rationally. There also should be conflicting interests of the participants. Obviously, earth would be an uninteresting place to dwell in if it was entirely free of conflict! The number of strategies available to each participant should be finite and may vary from participant to participant.

Here, a strategy refers to a complete plan of action a player will take, given the set of circumstances that might arise within the game. Corresponding to each combination of strategies, there is a pay-off (measured in terms of money or points) to the players. Let us assume that the strategy of a player dominates over the other strategies in all respects. Then, according to the principle of dominance, the latter strategies are ignored since they will have no effect on the solution in any way. If the strategy of a player, as the principle of dominance says, dominates over the other strategies in all respects, then the latter strategies are ignored since they will have no effect on the solution in any way. Determination of the superiority or inferiority of a strategy is based upon the objective of the player. Since each player is to select their best strategy, the inferior strategies can be eliminated.

Strategies can be either pure or mixed. Pure strategies are usually predetermined, while mixed strategies involve spontaneous decision-making on part of the player.

The games must also have certain rules and regulations that are known to the participants. The simplest type of game in game theory is perhaps the two-person zero-sum game [6]. This refers to a game where there are two participants with conflicting interests, and the net outcome for every possible combination of strategies is equal to zero. Each player has a finite number of strategies, and the gain of one player inevitably means the loss of the other (since they have conflicting interests).

Let us suppose that there are two players A and B. Player A has two strategies, while B has three. Thus, the possible combinations of strategies will be 3X2=6. Each combination provides a payoff for player A and a payoff for player B. Thus, we can form a payoff matrix.

Let A adopt $$i$$-th strategy and B adopt $$j$$-th strategy, where $$i=1,2$$ and $$j=1,2,3$$. Therefore, the payoff of player A is $$a_{ij}$$. With the use of different combinations of strategies of A and B, we can get the following payoff matrix:

$$ \begin{matrix} A\downarrow B\rightarrow & 1 & 2 & 3
1 & a_{11} & a_{12} & a_{13}
2 & a_{21} & a_{22} & a_{23}
\end{matrix} $$

According to the assumption, the sum of the payoffs of the two players A and B is zero. This means, when A receives $$a_{11}$$, B receives $$-a_{11}$$, and so on. In general, when A receives $$a_{ij}$$, B receives $$-a_{ij}$$.

Let us suppose that both A and B are tossing a coin separately. If both get the head, B is to pay $5 to A. If both get the tail, B is to pay $4 to A. In all other situations, A is to pay $3 to B. In this case, both A and B have two same strategies H and T. The gain of one is the loss of the other. Also, the sum of the payoffs of both the players must be zero.

The payoff matrix of A is:

$$\begin{matrix} A\downarrow B\rightarrow & H & T
H & 5 & -3
T & -3 & 4
\end{matrix}$$

When the strategy of A and B is (H,H), A gains $5. When the strategy of A and B is (T,T), A gains $4. For the other two combinations, the negative signs indicate A should pay $3 to B, or in other words, A faces a loss.

The pay-off matrix of B will be formed by taking the same values, but with opposite signs.

$$\begin{matrix} B\downarrow A\rightarrow & H & T
H & -5 & 3
T & 3 & -4
\end{matrix}$$

The optimum strategy of players in game theory is to maximize the gain or minimize the loss. In the maximin criteria, the maximizing player lists their minimum gains from each strategy and selects the strategy which gives the maximum out of these minimum gains. In the minimax criteria, the minimizing player lists their maximum loss from each strategy and selects the strategy which gives the minimum loss out of these maximum losses. This is known as the maximin-minimax principle.

A famous illustration in game theory is the prisoner’s dilemma [7]. Two prisoners are imprisoned for two years for a crime they committed together. They are separately offered a deal. To each prisoner, separately, a police officer says, “If you confess to the crime, and your partner does not, you will be freed from prison. However, in that case, your partner would serve ten years. If you both confess, you will both serve five years. If neither of you confess to the crime, you will both spend two years in jail.”

Now, if both keep their mouths shut, they only serve two years. That is the most ideal solution. However, they are asked to decide independently. Moreover, they are both criminals, and independent players in the game. They cannot trust each other completely. If one of them chooses not to confess in the hope that the other does not confess too, and they both serve only two years, then there is also the risk that in case the other confesses, the former would have to spend ten years in jail. Both the criminals have this same fear, and according to game theory, they both should confess and end up spending five years in jail. In light of the entire situation, this is the best solution. Both the players reach Nash equilibrium in this case, which simply means that they make the choice that leaves them relatively better off, irrespective of their opponent’s decision. Confessing leaves the player with two possibilities – either the prisoner escapes (if the opponent does not confess) which is the best possible result, else the prisoner serves five years (if the opponent confesses) which is still reasonable. The prisoner could choose not to confess with a blind faith in the opponent with the hope of getting out after two years. But that is an unstable state since the prisoner cannot trust the opponent and in case the opponent does confess, the prisoner would have to serve ten years. This thought experiment offers insight into how we unconsciously play strategic games all our lives, and demonstrates the importance of game theory in general, and not just as a tool used by economists.

The above scenario is an example of a non-cooperative game. An example of a cooperative game can be provided as follows [7]. Suppose person A can manufacture ten units in a certain period of time, alone. Person B can do twenty units alone, in the same amount of time. However, if they cooperate and work together, they can manage forty units in the same time, while individually it would have been thirty. They cooperate since both are to benefit. In such cases, game theory (more specifically, the Shapley value equation [8]) tells us how to determine the contribution of each individual to the overall outcome.

To determine person B’s marginal contribution to A, remove B from the game. A alone can make ten units, subtracting which from forty gives us thirty, which is B’s marginal contribution to A. In this case, A makes ten units. Similarly, B alone could make twenty units, meaning A’s marginal contribution to B is twenty units. Averaging A’s contribution in both cases, we get:

$$\frac{10+20}{2}=\frac{30}{2}=15$$

Thus, out of the forty units, A gets credit for fifteen, while B gets the credit for the rest twenty-five units. This makes sense. Out of the forty units, give A credit for the usual ten units, and B for the usual twenty units. That leaves us with ten units more. These ten units were produced only because A and B cooperated, so just divide them equally between A and B. Add five to both the initial values, and you see that A contributes fifteen and B twenty-five.

Game theory is a vast and unique field. As with all theories, problems remain with game theory. For instance, things like pay-off, number of players, etc., cannot always be predetermined. Also, game theory does not work for infinite strategies or competitors. Game theory assumes that the players are knowledgeable about the strategies available to their opponent, which in reality is not always the case. Also, the gain of a player does not always indicate a loss of another. All that said, game theory is a very promising and interesting field of study. It can tell us how to act smart in competitive situations, and also can help in our decision-making. Thus, indirectly, the concepts of game theory are applicable everywhere in our lives. Little doubt remains that game theory is one of the most important discoveries of modern times, and we are yet to understand the full implications of it. To quote George B Leonard, “In terms of the game theory, we might say the universe is so constituted as to maximize play. The best games are not those in which all goes smoothly and steadily toward a certain conclusion, but those in which the outcome is always in doubt.” [9]

References

[1] Paul Halpern. “Chaos Theory, The Butterfly Effect, And The Computer Glitch That Started It All”. 2018. https://www.forbes.com/sites/startswithabang/2018/02/13/chaos-theory-the-butterfly-effect-and-the-computer-glitch-that-started-it-all/#7374539869f6

[2] Don Ross. “Game Theory – Stanford Encyclopedia of Philosophy”. 1997. https://plato.stanford.edu/entries/game-theory/#:~:text=Academic%20Tools-,1.,and%20Oskar%20Morgenstern%20(1944).

[3] Lauren Cassani Davis. “Would You Pull the Trolley Switch? Does it Matter? – The lifespan of a thought experiment”. 2015. https://www.theatlantic.com/technology/archive/2015/10/trolley-problem-history-psychology-morality-driverless-cars/409732/

[4] Oskar Morgenstern. “Game theory – theoretical aspects”. 2020. https://www.encyclopedia.com/science-and-technology/mathematics/mathematics/theory-games#:~:text=The%20theory%20of%20games%20is,to%20determine%20the%20resulting%20equilibria.&text=Thus%2C%20in%20games%20of%20strategy,possible%20cooperation%20among%20the%20participants

[5] Daniel McNulty. “The Basics of Game Theory”. 2019. https://www.investopedia.com/articles/financial-theory/08/game-theory-basics.asp

[6] Saul I. Gass. “What is game theory and what are some of its applications?”. 2003. https://www.scientificamerican.com/article/what-is-game-theory-and-w/

[7] Hank Green (SciShow). “Game Theory: The Science of Decision-Making.” 2016. https://youtu.be/MHS-htjGgSY

[8] Marko Cotra. “Making Sense of Shapley Values”. 2019. https://towardsdatascience.com/making-sense-of-shapley-values-dc67a8e4c5e8

[9] George B. Leonard. “Some lines for you”. https://www.somelinesforyou.com/quote/652346

Figure References

Figure 1: “The Trolley Problem – Origins”. Accessed 25 September 2020. Medium. https://miro.medium.com/max/3550/1*8cs-6f1XCvrg1IIjTfk2zQ.jpeg

Figure 2: “Moral Reasoning & the ‘Trolley Problem’”. Medium. Accessed 25 September 2020. https://miro.medium.com/max/1012/0*thVUwrGurohPh7_y.jpg

About The Author

Arpan Dey, 16, is a high-school student from India who is deeply interested in physics, mathematics and the metaphysics of science. He has been working as an editor for the Young Scientists Journal and is involved in many STEM-related activities. He also enjoys reading, writing short stories (mainly sci-fi and detective) and listening to instrumental music. He is also an aviation enthusiast.

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