Game Theory

Six dimensions games can be categorize along

1. Sequential or simultaneous moves? 2.Are player's interests in total conflict or is there some commonality (zero, positive, or constant sum) 3.Is the game repeated or played once and do players change or stay the same? 4.Do players have full and equal information (do they know players, payoffs, rules,etc) 5.Are the rules and structure fixed or can they be manipulated? 6.Are agreements to cooperate enforceable?

Arrow's impossibility theorem

1. The social or group ranking must rank all alternatives (be complete). 2. It must be transitive. 3. It should satisfy a condition known as positive responsiveness, or the Pareto property. Given two alternatives, A and B, if the electorate unanimously prefers A to B, then the aggregate ranking should place A above B. 4. The ranking must not be imposed by external considerations (such as customs) independent of the preferences of individual members of the society. 5. It must not be dictatorial—no single voter should determine the group ranking. 6. And it should be independent of irrelevant alternatives; that is, no change in the set of candidates (addition to or subtraction from) should change the rankings of the unaffected candidates.

mix strategy payoff

1.Find probability of each outcome in equilibrium 2. multiply that probability by one player's payoff for each outcome 3.sum all of those numbers

antiplurality method,

Asks voters to vote against one of the available alternatives or, equivalently, to vote for all but one. For counting purposes, the alternative voted against is allocated 1 point, or else all alternatives except that one receive 1 point while the alternative voted against receives 0.

Strategic form

In game theory, the strategic form (or normal form) is a way of describing a game using a matrix.

Prospect Theory

Prospect theory is a behavioral economic theory that describes the way people choose between probabilistic alternatives that involve risk, where the probabilities of outcomes are unknown. The theory states that people make decisions based on the potential value of losses and gains rather than the final outcome, and that people evaluate these losses and gains using certain heuristics. (Gain v Loss Frame) where people are more risk adverse when their are in the gain frame so they go for a sure thing (x people will be saved). In the loss frame they go for the risk (x people will die)

winner's curse

The winner's curse is a tendency for the winning bid in an auction to exceed the intrinsic value or true worth of an item. Because of incomplete information, emotions or any other number of factors regarding the item being auctioned, bidders can have a difficult time determining the item's intrinsic value.

Prisoner's Dilemma

There is a better outcome if they cooperate but they will go to the nash equilibrium which is defect defect

Chicken Paradigm

This story has four essential features that define any game of chicken. First, each player has one strategy that is the "tough" strategy and one that is the "weak" strategy. Second, there are two pure-strategy Nash equilibria. These are the outcomes in which exactly one of the players is chicken, or weak. Third, each player strictly prefers that equilibrium in which the other player chooses chicken, or weak. Fourth, the payoffs when both players are tough are very bad for both players. In games such as this one, the real game becomes a test of how to achieve one's preferred equilibrium.

Assurance Paradigm

Two Nash Equilibria. One is prefers by both players. They must be assured the other will choose the appropriate action.

Pure Coordination Paradigm

Two Nash Equilibrium. Both have the same payoffs.It is important for both that they achieve one of the equilibria, but which one is immaterial because the two yield equal payoffs. All that matters is that they coordinate on the same action; it does not matter which action.

Battle of the Sexes Paradigm

Two nash equilibria. Bother preferred to a different player.

predo-optimal choice

add both numbers and the highest collective value is the predo-optimal choice (the collectively best option)

single transferable vote method (instant runnoff)

can be used to tally votes in later rounds. With a single transferable vote, each voter indicates her preference by ordering all candidates on a single initial ballot. If no alternative receives a majority of all first-place votes, the bottom-ranked alternative is eliminated and all first-place votes for that candidate are "transferred" to the candidate listed second on those ballots;

Stag Hunt

describes a conflict between safety and social cooperation (social cooperation problem)

Equilibrium

each player is using a strategy that is best response to strategies used by others

majority runoff procedure

for instance, is a two-stage method used to decrease a large group of possibilities to a binary decision. In a first-stage election, voters indicate their most-preferred alternative, and these votes are tallied. If one candidate receives a majority of votes in the first stage, she wins. However, if there is no majority choice, a second-stage election pits the two most-preferred alternatives against each other. Majority rule chooses the winner in the second stage.

The Condorcet paradox

is one of the most famous and important of the voting paradoxes.5 As mentioned earlier, the Condorcet method calls for the winner to be the candidate who gains a majority of votes in each round of a round-robin of pairwise comparisons. The paradox arises when no Condorcet winner emerges from this process. (intransitive ordering)

pairwise voting

multistage binary vote where two people are paired against each other

rationality

order outcomes, act to maximuze value

payoffs

outcome values and preferences

how to solve for p

p(first across payoff)+(1-p)(second across payoff) = p(first down payoff)+(1-p)(second down payoff)

plurality rule

person with the most votes wins and doesnt need majority

subgame perfection

rollback equilibrium reached by doing optimal thing at each stem

Condorcet method

round robin style pairwise vote

common knowledge of rules

rules are list of players, availible stragigies, payoffs to each player for each stratigy configuration

The Borda count

voters rank options or candidates in order of preference. The Borda count determines the outcome of a debate or the winner of an election by giving each candidate, for each ballot, a number of points corresponding to the number of candidates ranked lower. Once all votes have been counted the option or candidate with the most points is the winner. Because it tends to elect broadly-acceptable options or candidates, rather than those preferred by a majority, the Borda count is often described as a consensus-based voting system rather than a majoritarian one

Reversal Paradox

when borda count is used and one of the candidates is removed

approval voting method,

which allows voters to cast a single vote for each alternative of which they "approve."