The political arena is defined by such decisions on conflict and cooperation involving civilians, political parties, religious groups, social movements, corporations, and even revolutionary groups. The process with which one can arrive at a rational, well-thought-of decision in politics seems to be overwhelming, given the complexity of individual and social behavior. Several variables have to analyzed, and possible scenarios have to be completely scanned. To make things worse, the cost of wrong decisions can take a toll on lives, properties, and positioning. With all the complexity involved, is there a tool one can actually use to make political decision making easier?
Here comes "Game Theory" - a subfield in applied mathematics that deals with modelling "strategic situations" i.e. situations wherein an individual's success in making choices depends on the choices of others (Myerson, 1991). Emerging from the field of economics, game theory has been increasingly applied to analysis of political situations - gaining prominence during the Cold War between the United States and Soviet Union.
But game theory's application is not just limited in political science or economics. In fact, social morality and ethics may have evolved from social conventions that are, as will be explained later, "Nash equilibrium". For starters, check out the Saturday Morning Breakfast Cereal (SMBC) comic below:
The question then becomes, what would people decide to do eventually? Can we really predict what they would decide to do given the respective payoffs of their decisions? These questions are those that are answered in the field of game theory. At its very core, game theory assumes that people are "rational", and so they would consistently select strategy that will optimize their benefits. From this assumption, we can now set up a "game" based on what we know about people's respective benefits as assigned to their respective choices relative to the choices of others. From there, we can then predict players' best strategies and assume that, since they are rational, this will be their probable behavior in the real world.
The Nash Equilibrium and Social Convention
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| John Forbes Nash, Jr. |
Imagine that in the game, everyone knows what's the "best" strategies of each other players are. We can take "best" to mean that it is the strategy with the best possible outcome given all response or moves of any other players. These best strategies are called "Nash strategies". Now, if none of the player has anything to gain by changing his own strategy unilaterally (and others stick to their respective strategies), then the game is said to be in "Nash equilibrium". This means that no one has any interest in changing any of the move because they will become worse off by doing so.
Let us take a look at the SMBC comic. Consider a situation wherein Prisoners 1 and 2 are in the fourth quadrant i.e. they both decide to pick the strategy of remaining silent. Each then will have 6 months. However, if one decides to rat out on the other, he has something to gain. There is an incentive for one to change strategies; actually both of them have an incentive to do so. Thus, quadrant four (D) is not a Nash equilibrium. Checking out each quadrant we'll discover that only quadrant one (A) is a Nash equilibrium - no one of them stands to gain by changing strategies i.e. by remaining silent.
Only the Nash equilibrium is a "stable" result of a game - that is, once players converge to their Nash equilibrium, we can expect them to stay there. In fact, in most social situations, "social convention" can actually be seen as the Nash equilibrium of that particular social game. Norms, practices, traditions are created to preserve the equilibrium, and thus, social stability. Incentives and disincentives, like social acclaim or censure, make it costly for any individual to deviate from the convention. You decided to skip a reunion to watch a movie? You'll likely be ostracized in the next. The United States skipped on giving "aid" to North Korea? Whoa, another nuclear crisis. A Philippine President attempts charter change near the end of her term? She'll have to face a united wall of opposition from all other former Presidents.
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| Sheldon Cooper from the Big Bang Theory. Picture from here. |
Deviations and Mixed Strategies
But how do we explain the persistence of stability with periodic or even random deviations? Here, we introduce two more concepts: mixed strategies and repeated games. Let's deal with mixed strategies first.
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| Figure 1. Payoff matrix for Rock Paper Scissors |
But what is a players' best strategy? The point of a best strategy actually is to achieve the best result possible rather than the best possible result. In the rock-paper-scissors, for instance, the aim is to out-score your opponent, not to win at every turn. With this notion, a player may then begin mixing his strategies so as to maximize the probability of him winning. How will he then do that?
This element is simply not present in a rock-paper-scissors game as presented above because the payoff for any victory is equal as the chances of your opponent to pick any of the three alternatives are equal. But consider the 1998 findings by Japanese mathematician Mitsui Yoshizawa that in a study involving, 725 people, rocks were thrown 35% of the time, paper 33%, and scissors 31%. In this case, a player playing mixed strategy would go papers 35%, scissors 33%, and rocks 31%. This is his "mixed Nash strategy". If all the players play their mixed Nash strategies, then we have mixed Nash equilibrium.
To learn more about Mixed Nash equilibrium, one can visit gametheory101.com. It contains instructional videos on game theory. The one specifically on mixed strategy is shown below:
So how are mixed Nash strategies and equilibrium relevant to our notion of "social convention"? It actually explains a lot. With individuals knowing the cost and benefits of deviation and non-deviation can then mix his or he strategy. With all of the people doing so, we can then scan society and pick a person in random, the probability that he is pursuing a particular type of strategy is related to the payoff of that strategy. We then have a stable, Nash equilibrium, society amid the persistence of the deviants.
In politics, this means a lot (you may want to check out this link). Being unpredictable is one the most useful traits a politician can have - the element of surprise is as important a trait in politics as in war. Almost all great political leaders have been unreadable in their intentions, which remaining rational and committed to a goal. But there are contemporary examples, like in the rise of Juniricho Koizumi.
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| Permanent link to this comic: http://xkcd.com/661/ |
Pareto Optimality and Risk Dominance
It seems that Nash equilibrium, mixed or perfect is enough to create stability of choices among players in a game. But is stability good in itself? Consider, for instance the "Do unto others" game above. "Like-Like" is the Nash equilibrium, it is the most stable outcome. However, it is hardly desirable. The most desirable outcome is actually not a Nash equilibrium: quadrant D with "would have them do unto you-would have them do unto you".Here is where we introduce to concept of Pareto Optimality. We use the definition from GameTheory.net:
Named after Vilfredo Pareto, Pareto optimality is a measure of efficiency. An outcome of a game is Pareto optimal if there is no other outcome that makes every player at least as well off and at least one player strictly better off. That is, a Pareto Optimal outcome cannot be improved upon without hurting at least one player. Often, a Nash Equilibrium is not Pareto Optimal implying that the players' payoffs can all be increased.On the Prisoners' Dilemma game, the Nash Equilibrium "Rat Out-Rat Out" can actually be improved upon i.e. we can increase the payoff of one player without hurting the other (in fact, we will notice that the other player's payoff will improve). On this game, "Remain Silent-Remain Silent" is in fact the Pareto Optimal, even if it is not a Nash Equilibrium.
What is the importance of Pareto Optimality? Actually, it can serve as a guide for us in choosing among multiple Nash equilibria (in both mixed and perfect case). Let us then consider another game - the "Stag Hunt" game which was based on Jean-Jacques Rousseau's thoughts on social cooperation. Imagine two players going out on a hunting trip, and each has an option of pursuing a stag - a more satisfying catch that can only be caught by two men, or a rabbit - a smaller catch but can be captured by only one person. Thus, if one decides to hunt for a stag, and the other a rabbit, then the one who went stag hunting will eat nothing, and the one who hunted rabbit will be able to eat a rabbit. Stag hunting is a risky enterprise at this note.  |
| Figure 2. Payoff Matrix for the Stag Hunt Game |
However, even if one knows that Stag-Stag is both a Pareto Optimal and a Nash equilibria, there is a chance that the other may pursue the other Nash equilibria, Rabbit-Rabbit. After all, by choosing rabbit, you are guaranteed a payoff of 3, whether the other pursues a rabbit or a stag. For this reason, the rabbit equilibrium is a risk dominant one - which is similar to saying that it is the "safe option". The stag equilibrium is the payoff dominant one, because it is Pareto optimal. Cooperation between members two players gives a good chance of success in a risky venture, but an individual can win a guaranteed but lower reward by breaking the cooperation and going it alone.
What did we learn here? First, Nash equilibria are not necessarily Pareto optimal - thus stable social conventions are not necessarily the most efficient for all. Real life politics and economics are abound by examples of this. The recent debt deal in the US, the collapse of the six-party talks in North Korea, the failure to realize a two-state solution for the Israel-Palestine conflict, among many other instances of compromising too much.
Social Dilemmas
The Stag Hunt game and the Prisoners' Dilemma are just two of the social games one may find itself into. Actually, there are many others. An excellent book on elaborating on this this is "Game Theory in Everyday Life" by Len Fisher, a physicist (there is a nice interview of him here). In Chapter 3 of his book, Fisher sketches some of the social dilemmas one may face, and interpreting it using the lens of game theory.
The other social dilemmas are (directly lifting from Fisher, italicized text, links mine):
- The Tragedy of the Commons, which is logically equivalent to a series of Prisoner’s Dilemmas played out between different pairs of people in a group.
(There is an opposite phenomena. The tragedy of the anticommons occurs when the existence of numerous rightsholders frustrates socially desirable outcome. As such, too many property rights could lead to less innovation. It mirrors the older tragedy of the commons used to describe problems arising from insufficient rightsholders. )
- The Free Rider problem (a variant of the Tragedy of the Commons), which arises when people take advantage of a community resource without contributing to it.
- Chicken (also known as Brinkmanship), in which each side tries to push the other as close to the edge as they can, with each hoping that the other will back down first. It can arise in situations ranging from someone trying to push into a line of traffic to confrontations between nations that could lead to war, and that sometimes do.
- The Volunteer’s Dilemma, in which someone must make a sacrifice on behalf of the group, but if no one does, then everyone loses out. Each person hopes that someone else will be the one to make the sacrifice, which could be as trivial as making the effort to put the garbage out or as dramatic as one person sacrificing his or her life to save others.
- The Battle of the Sexes, in which two people have different preferences, such as a husband who wants to go to a ball game while his wife would prefer to go to a movie. The catch is that each would rather share the other’s company than pursue their own preference alone.
Emerging from Economics
Originating from economics, game theory has been reinventing economics itself. In one of the lectures in our microeconomics class, our professor Dr. Michael Alba said that in the next decade or so, economists will be investing less on classical theory of demand (and its underpinning "theory of convexity" to guide its drive for "optimal solutions") and more and more on Game Theory. And this is understandable, given that in mainstream economics, competition and cooperation (in consumption and production) are the primary determinants of welfare, and the choice of whether to compete and cooperate and with whom often entails thinking of what will other people's behavior be, and how will that affect the outcome of your decision.
This means that even the mainstream believes now that it is not enough anymore to assume Adam Smith's "invisible hand" will put us all in good shape as long as we pursue our own economic interests. In fact, there is a set of social choices which, while individually optimal - will result to less than socially optimal results. Game theory - as a powerful tool for modeling social choice - is used to depict and explain such situations. Moreover, since social choices are not just economic in nature (we can have political, cultural, personal decisions), game theory is a powerful tool that can be used to explain phenomenon such as morality and ethics.
To follow: Credibility in Repeated Games
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