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Game theory is a set of tools used to help analyze situations where an individual’s best course of action depends on what others do or are expected to do. Game theory allows us to understand how people act in situations where they are interconnected.
Game theory is useful whenever there is strategic interaction, whenever how well you do depends on the actions of others as well as your own choices.
Game theory is the study of strategic interaction. Strategic interaction is also the key element of most board games, which is where it gets its name. Your decision affects the other player’s actions and vice versa. Much of the jargon of game theory is borrowed directly from games. The decision makers are called players. Players make a move when they make a decision.
Real-world strategic interaction can be very complicated. In human interaction, for instance, it’s not just our decisions, but also our expressions, our tone of voice and our body language that influence others.
backward induction: you can figure out your opponent’s response to your possible actions and take that into consideration before making your own move.
Human behaviour is probably better approximated by bounded rationality. That is, human rationality is limited by the tractability of the decision problem (how easy it is to manage), the cognitive limitations of our minds, the time available in which to make the decision, and how important the decision is to us.
Often players do not know the actions of the other players when they make their own decisions. Games with this feature are called simultaneous-move games. In some cases the players are literally making their choices simultaneously (at the same time). In other cases, they might be making their choices at different times. But as long as they do not know what action the other players have chosen at the time they make their own decisions, we can treat them as moving simultaneously.
The payoff numbers in the table represent the total payoff that the players get from each outcome: in a particular outcome a player might benefit directly, as well as indirectly from hurting or helping others. The payoff numbers include everything they care about.
The idea of Nash equilibrium is both simple and powerful: in equilibrium each rational player chooses his or her best response to the choice of the other player. That is, he or she chooses the best action given what the other player is doing.
The game illustrates the difficulty of acting together for common or mutual benefit given that people pursue self-interest.
Prisioners' Paradox
Notice that in putting the game into strategic form we do not say anything about what is likely to happen. We simply put down all potential outcomes, whether reasonable or not, and record the payoffs the players would get if that outcome occurred.
An outcome is Pareto efficient if there is no other potential outcome where somebody is better off and nobody is worse off. This notion of distributional efficiency is named after the Italian economist Vilfredo Pareto (1848–1923).
The Nash equilibrium outcome of the Prisoners’ Dilemma is not Pareto efficient because each prisoner would have been better off if both had remained silent; hence the nickname “Prisoners’ Dilemma”.
For example, when wireless network routers, such as Wi-Fi routers or cell-phone towers, use the same frequency and are within range of each other, they interfere with each other’s communication, slowing down the speed of both routers. One solution to this problem is to lower the transmission power of both routers so that they are no longer within range of one another. But if only one router is using low power, then its signal is overwhelmed by the high-power router.
In the economics literature, the term “commons” has evolved to encompass any shared resource.
Each country must decide whether to increase its nuclear arsenal. If a country doesn’t add to its arsenal, it saves the cost and implicit risk of accident. But each country has an incentive to increase its arsenal to enhance its geopolitical position. It is in each country’s self-interest to invest in nuclear weapons, no matter what the other county is doing. The Nash equilibrium of the game, therefore, is global nuclear build-up.
In the Prisoners’ Dilemma, although there is benefit to cooperative behaviour, individual incentives encourage conflict. In the network engineering example, it is possible to overcome this problem if one person controls both routers. But in human interaction achieving cooperation can be more difficult.
There is a free-rider problem in the Roommate Game. Alice has the highest payoff when she relaxes while Beth does the dishes. The same goes for Beth.
International cooperation on environmental protection is like the Roommate Game writ large. Each country prefers to remain passive while the others adopt costly abatement technologies to reduce CO2 emissions.
So far we’ve looked at games with a single Nash equilibrium. In these games, the Nash equilibrium gives a single prediction for the players’ behaviour.
It is plausible that the couple in the Battle of Sexes Game ends up with coordination failure due to misaligned expectations. In this case the game theorist would observe an “out-of-equilibrium” outcome where the couple spends the evening separated: neither of the two possible Nash equilibria come to pass.
In environments with multiple equilibria, players may coordinate their expectation on one equilibrium using social norms.
In games with multiple equilibria, if a social norm is not present, players may use a coordination device, a shared observation or common history to help coordinate expectations on the same equilibrium.
Hence, no matter how healthy the financial state of a bank may be, any bank will go under if faced with a bank run (where everybody tries to withdraw their money at the same time).
As in the Battle of the Sexes Game, in banking there are multiple Nash equilibria. Depending on people’s expectations, we may observe business as usual or a bank run.
The belief that there will be a bank run is a self-fulfilling expectation: the expectation itself causes the bank run.
Even positive statements or actions by bankers or policymakers can backfire if people take them as a sign of weakness.
So far we have looked at games that have a pure-strategy Nash equilibrium. This is an equilibrium in which players pick a particular choice with certainty.
The Rock-Paper-Scissors Game has just one equilibrium: each player plays a mixed strategy of choosing each of the three possible choices, (rock, paper or scissors) with equal probability.
Mixed-strategy Nash equilibrium has applications in a wide variety of fields. It can capture the spirit of surprise in games where players are unpredictable.
Harsanyi points out that even if players play pure strategies, if they are slightly uncertain about each other’s payoffs, from the outside they will seem as if they are randomizing between actions.
To understand when players collude in a Prisoners’ Dilemma type of situation, we need to move beyond one-shot games (where players play the game only once and then the game ends) and to start thinking about more realistic settings with repeated interaction, where players play the same game again and again.
Imagine that both players know that they will play the Prisoners’ Dilemma Game not once but twice. To find the equilibrium of the game with repeated interaction, we first predict the equilibrium of the game in the last round. And then we reason what the equilibrium would be in the first round. This line of reasoning is called backward induction.
Players can reason that there will be no cooperation in the second round no matter what happens in the first round. Hence, from the players’ viewpoint, the first round of the game is no different from a one-shot Prisoners’ Dilemma either. So, in equilibrium there is no cooperation at any stage of the game.
Evolutionary forces do not necessarily lead to the best outcome for a species. Competition for scarce resources often means that individual benefits and group benefits are in opposition.
The tension between group benefits and individual benefits is present in physical traits as well as in behavioural patterns.
One brake on this evolutionary process is the rise of a rival species which contests the same ecological resources: if a species becomes too inefficient due to its large size, it will eventually be pushed out by a more efficient competing species.
evolutionarily stable equilibrium. It is an equilibrium which is stable in the sense that if we add a small number of animals with different conditioning, evolutionary forces will eventually restore the equilibrium.
Hawk-Dove Game, there is a single evolutionarily stable equilibrium and the long-run steady state will eventually be restored regardless of how many animals with different conditioning we add.
But some games have more than one evolutionarily stable equilibrium. In these games, evolutionary forces will restore the equilibrium proportions if there are small changes to the population. But large changes in the population composition can cause evolutionary forces to bring the population to another equilibrium altogether.
Oddly enough, the evolutionarily stable proportion of hawk-types (5/6) is also equal to the equilibrium probability in the mixed-strategy Nash equilibrium of the game if the animals were choosing their strategies rationally. This is not a coincidence. To calculate the equilibrium probabilities in the mixed-strategy Nash equilibrium, we look for probabilities where players are just indifferent between the hawk and dove strategies. In equilibrium their expected values from both strategies are equal.
In an evolutionary environment, focusing on the evolutionarily stable equilibria is a reasonable way to rule out equilibria which could not survive even small changes to the underlying population.
the extensive-form representation. This is also called the game tree.
decision nodes, dots that represent a point at which a decision can be made.
This is a subgame-perfect Nash equilibrium: players have best responses to each other in each subgame of the original game. Subgame perfection implies that players are forward-looking. They do the best they can at each decision node they encounter without either holding grudges or developing goodwill for past actions.
Subgame perfection discards Nash equilibria that depend on players making non-credible threats and non-credible promises.
This is called the time-inconsistency problem: the decision maker does not find it optimal to follow the original action plan.
Financial markets often use collateral as a commitment device. For instance, the applicant could use her family home as collateral. As long as losing the family home would be costly enough for the applicant (either financially and/or in psychological suffering), the collateral changes the expected payoff of the risky project to the applicant. So, she would choose to invest in the safe project. Hence the bank would approve the loan.
However, those who do not have existing assets to use as collateral will have their loan applications rejected in the subgame-perfect Nash equilibrium because of the time-inconsistency problem. Due to the difficulty of coming up with a commitment device, the poor stay poor, while the rich get richer. Lack of access to credit markets can prevent poor people from being upwardly mobile, which can cause severe social unrest and violence.
