Unit 4 Flashcards
What are social interactions and how can they be modelled?
Social interactions are situations in wich there are two or more people, and the actions taken by each person affect both their own and others’ outcomes.
Game theory can be used to model how people interact in strategic interactions by modelling them as games.
Explain how a social interaction can be modelled as a game.
The game specifies:
The players - who is interacting with whom
The feasible strategies - which actions are open to players
The order of play - when the players can choose their actions
The information - what each players knows when making a decision
The pay-offs - what the outcomes for each possible combination of actions will be
What is a payoff matrix, Nash equilibrium and dominant strategy and dominant strategy equilibrium?
A payoff matrix is a rectangular array of numbers showing the possible outcomes for both players in a game.
A Nash equilibrium is an outcome where both players have played their best response to the strategies of the other players.
A dominant strategy is an action that gives a player the highest payoff, regardless of what the other players chooses.
A dominant strategy equilibrium is an outcome whereby both players have played their dominant strategy.
What is the Pareto criterion?
According to the Pareto criterion, allocation A is better than allocation B if at least one party would be strictly better off with A than B, and nobody would be worse off. A Pareto-dominates B.
What is Pareto efficiency?
If an allocation is Pareto efficient then there is no alternative allocation in which at least one party would be better off and nobody worse off. The allocation is not Pareto dominated by any other allocation. Often games can have multiple Pareto efficient outcomes.
What is a prisonner’s dilemma game and what is the outcome?
A game in which the players’ dominant strategy equilibrium is not Pareto efficient. This means by cooperating, both players could reach an outcome in which they both receive a higher payoff off, but because this isn’t their dominant strategy and it doesn’t Pareto dominate all the others, this outcome is never reached without cooperation.
What is an invisible hand game and what is the outcome?
A game in which the Nash equilibrium is Pareto efficient and Pareto dominates all other allocations. Both players play their best response and end up in a Nash equilibrium.
What is a public good game and what are the outcomes?
A game in which individual players can take an action that would be costly to themselves but would produce benefits for all players including themselves.
For each player, they would receive a higher payoff by not contributing and free riding on the contributions of others.
If each player only cares about their own payoff, the dominant strategy is to not contribute and all payoffs are zero.
Thus this is a form of prisonner’s dilemma game whereby by cooperating, each player would receive a higher payoff leading to a Pareto efficient outcome but this does not happen as each player would do better by free riding on others.
What are some real world examples of the public goods game?
The public goods game comes into play in many real life irrigation projects whereby farmers or fishermen can choose to invest in projects that would benefit all farmers/fishermen. Farmers can choose to invest, where collective investment would benefit everyone in the form of significant improvements in capital.
Farmers can choose to not invest and free ride on the investments of others, but run the risk of everyone choosing not to invest, leading to zero investment and no payoffs.
How does altruism encourage players to choose cooperative outcomes and deter from choosing their dominant strategy?
Some players in a game may have social preferences, when their individual utility depends their own pay off and what happens to other people. Altruism is a social preference i which a. Individual’s utility is increased by benefits to others.
In an example where player A can choose how they split £X between themselves and another player, how can indifference curves be used to model this decision?
If player A is self interested, their indifference curves (with A’s income on the x axis and B’s income on the y axis) would be vertical lines. Increases in their own income increase their utility, but the income to player B has no effect on their utility.
If player A is altruistic, their indifference curves would be downward sloping curves, showing that they are willing to give up some of their own income to give more to player B.
The more altruistic player A is, the flatter the indifference curves, as player A is willing to give up more income to give more to B.
By drawing player A’s feasible set (where the combined income of both players = 200 or less), A’s decision will be where an indifference curve just touches the feasible set.
How do repeated games affect decisions of players?
If a game is to be repeated, players may have a different best response than to simply maximise their own payoff, given that acquiring a reputation of being uncooperative may lower their future payoffs.
When playing a repeated public good’s game, ie where players can choose to contribute and there are several rounds of play where players can observe how much everyone is contributing, what normally plays out?
Initially, players contribute around half their money into the common pool. As rounds go on, the average contribution becomes lower and tends to zero. This can be explained by social norms, whereby players initially contributed as they saw others were doing so, but as some players started to free ride, other players began to lower their contributions in response as a form of punishing the lack of reciprocity.
What is the ultimatum game?
The ultimatum game is a one-shot, sequential, take-it-or-leave-it game played by 2 players. First, player 1 (the proposer) is given $100 and proposes a split of the money between the two players. Player 2 (the responder) then chooses to accept or reject the offer. If the offer is accepted, the players walk away with their respective amounts decided by the proposer, but if rejected, both players walk away with nothing.
What are some likely outcomes of the ultimatum game?
The outcomes depend on the players’ social preferences and their attitudes to reciprocity and social norms.
A responder with purely self-interested preferences would accept any positive offer as something is better than nothing.
If the proposer too were self-interested, they would offer the minimum amount, $1, to the responder, knowing it would be accepted. This is a Pareto efficient but unfair outcome.
If the proposer cared about fairness, they may propose a 50-50 split.
Even a selfish proposer may expect the responder to reject a low offer and so the proposer needs to propose the minimum acceptable offer - the lowest amount that would be accepted - in order to not walk away with nothing.
What is the responder’s minimum acceptable offer in the ultimatum game?
Assume the social norm is a 50-50 split.
The minimum acceptable offer is the amount where the pleasure of receiving the money=the satisfaction of rejecting.
When the offer, y, is $50 or more, the responder accepts, as rejecting would hurt them and the proposer which they have no intention of doing given that the proposer has conformed to the social norm.
If y is lower than 50, the responder rejects as they feel the social norm is not being respected and may want to punish the proposer.
To model this reciprocity motive, the responder’s satisfaction from rejecting = R(50-y)
R is a number that measures the strength of the reciprocity motive, the larger R is, the more they care about i the proposer is acting generously or not.
The responder should only reject the offer if y is lower than the satisfaction of rejecting y. Ie y<R(50-y)
Rearranging this gives, reject y if y< 50R/1+R and this is the minimum acceptable offer
What is a coordination game?
A game in which there are two Nash equilibria, one of which may Pareto dominate the other. Each player would like to ensure their action coordinates with their opponents action.
How can game theory be used to explain why it is difficult to agree international climate action?
Using payoff matrixes to model the outcomes between china and USA if their only options were to reduce pollution or continue pollution, we see that the Pareto efficient outcome is for both to reduce but this is not an equilibrium. The best outcome for each country is to continue whilst the other reduces, thus free riding. The outcome where both continue is better for each country than to reduce whilst the other continues.
In a prisonner’s dilemma game, continuing pollution is the dominant strategy for both countries.
In a coordination game, both countries would like to coordinate on the same action. The best outcome would be for both to reduce, but in fear of the other continuing, we can expect both to continue.
