Games Markets & Information Flashcards
L1 - What are the 4 features of a normal form game, with notation?
Players in the game N = {1, 2, …, n}
A set of pure Strategies available to each player {S1, S2, …,Sn}
A set of Payoff Functions for each player for each combination of strategies that could be chosen by the players {π1, π2, …, πn}
This leads to a normal-form game being a triple of sets: {N, {Si}^n, {πi (.)}^n}}
Information
L1 - Normal Form Game Feature definition: Players?
Are individuals who make decisions.
L1 - Normal Form game Feature definition: Actions?
An action by player i is a choice they can make. Player i’s action set is the entire set of actions available to i.
L1 - Normal Form game Feature definition: Payoffs?
By payoff, we mean the utility player i receives after all players have picked their strategies and the game has been played out.
L1 - Normal Form game Feature definition: Information?
Specifies what player I knows at different points in time. Usually modelled as information sets.
L1 - Strategy in the terms of games?
A plan of action intended to accomplish a specific goal. A Player i’s strategy s(i) is a rule that tells him which action to choose at each instant
of the game, given his information set.
L1 - For a Set of s* = s1, s2, …, sn, what does each s represent? What is the definition of the s equilibrium?
A strategy that can be undertaken by each i player. Each player will have a on best strategy (s) of this set. So an equilibrium s* is an ordered set consisting of one best strategy for each of the n players of the game.
L1 - What are the three sets we use in a normal form game and their notation?
A set of players N = (1, 2,…, n)
A set of pure strategies S = (S1, S2, …, Sn)
A set of payoffs, V = (π1, π2, …, πn).
L1 - How would you represent the payoffs for each player in the prisoners dilemma using the symbols in the slides?
[π1(c,c), π2(c,c)]
[π1(s,c), π2(c,s)]
[π1(c,s), π2(s,c)]
[π1(s,s), π2(s,s)]
L1 - How would we denote (symbols) of a normal form representation game?
An n-player game with the strategy spaces S1, …, Sn and their payoff functions π1, …, πn. We denote this game by G = {S1, …, Sn, π1, …, πn}
What does the payoff matrix represent?
The utilities of each player given the strategies taken by each player.
L1 - What is the table called that represents the utilities of N players given their strategies?
The payoff matrix.
L1 - What is the difference between Si and si?
Si is the list of potential strategies that player I could choose to take. si is the choice the strategy that I chose to take, which will be an element of Si.
L1 - S-i?
The list of strategy sets of all players excluding player i.
L1 - What is the payoff notation for players i’s best response (s*i) given all the other platers responses?
πi (si, s-i) Is greater than or equal to πi(s’i, s-i) for an S’i not equal to si.
L1 - The difference between strongly (strictly) best response and weakly best response?
Strongly (Strictly) best response is the best response if no other strategies are equally as good given the other players strategies, whereas the weakly best response occurs when another strategy can be equally as good.
L1 - When is strategy s*I a dominant strategy? Define and denote.
If it is a players strictly best response to any strategies the other player might pick, in the sense that whatever strategies they pick, his payoff is highest with si.
πi (si, s-i) Is greater than πi(s’i, s-i) for an S’i not equal to s*I and any S-i.
His inferior strategies are dominated strategies.
More clearly, a dominant stratify is the best stratify or a player no matter what strategy the other player chooses.
L1 - What is a dominant strategy equilibrium?
A strategy combination consisting of each player’s dominant strategy. It occurs when each player has a dominant strategy, so each player will choose this strategy leading to an equilibrium.Both
L1 - How would we denote the Best Response of a player 1, given player 2 is choosing to confess, if that would lead player one’s best response to be to confess?
BR(S) = C
L1 - Assumption:
1. All players are rational.
2. All Players knows the game/ interaction.
Claim: A rational player will never play a strictly dominated strategy.
Knowledge of the game implies that a player should recognise dominated strategies and rationality implies that these strategies should be avoided. Remember, it is assumed that the players do not know each others payoff matrix. Given the definition of the dominated strategy, this seems straight forward.
L1 - What are strictly dominated strategies and what is the assumption?
We say that s’i is strictly dominated by si if for any possible combination of the other players’ strategies,
si is an element of Si, player i’s payoff from s’i is strictly less than for that of si. See notes or p.60 of text book for further information and the definition.
L1 - Why might the Dominant Strategy Equilibrium fail?
Some games do not have a dominant strategy for players, and thus we do not attain a dominant strategy equilibrium.
L1 - How can we link in the Pareto Criterion into the solution concept (strictly dominant equilibrium) and what would the implication be?
It is often the case that a strictly dominant equilibrium will not be Pareto optimal. It implies that a player could modify the environment by creating some ENFORCEMENT MECHANISM to attain the best result.
L1 - Enforcement Mechanism?
Introduced with regards to the Pareto criterion with reference to a dominant equilibrium. An enforcement equilibrium can be implemented to modify the environment e.g. a mafia.
L1 - Weak Domination?
In any game, a players actions “weakly dominates” another action if the first action is at least as good as the second action, no matter what the other players do, and is better than the second action for some actions of the other players.
See Brainscape Assistant 1 for an example.
See P.63 of book or slide 17/22 of Lecture 1.
L1 - So what are the differences between a dominant and weakly dominant strategy?
A dominant strategy requires the strict best response, the players cannot be indifferent between the two choices. A weakly dominant strategy allows for indifference.
L1 - Pure strategy notation:
profile of pure strategies and set of all strategies.?
The set of all pure strategies for player i is denoted Si. A profile of pure strategies, s =(s1,s2,…, sn), si ∈ Si for all i = 1,2,…,n, describes a particular combination of pure strategies chosen by all n players in the game.
Si represents the set of all strategies that i can undertake.
L1 - What do we mean by ‘pure’ in pure strategies?
Players choose deterministic action, namely that players choose a certain plan of action.
L1 - A collection of sets of pure strategies: notation?
{S1, S2, …, Sn}, where each player will simultaneously choose a possible strategy such that si ε Si.
L1 - What is a Cournot Duopoly?
A variant of
Qualities
Behaviour
COURNOT : QUANTITIES
A variant of the prisoner dilemma in which two identical firms produce some good.
We assume there are no fixed costs of production and that the variable cost to each firm is i of producing quality qi ≥ 0.
Firms choose quantities and the market price adjusts to clear the demand.
L1 - A finite game?
A game with a finite number of players, in which the number of strategies in Si is finite for all players i ε N.
L1 - When would a payoff matrix not be able to be used?
When the game is not finite.
L1 - What notation do we introduce to enable us to refer to the an opponents strategy set?
We previously define a player profile as s = s1, s2, …, s(i-1), s(i), s(i+1), …, sn) where si ε Si as πi(s). This represents the strategies taken by all players (n).
s(-i) represents player i’s best response to the strategies of all the other players, s = s1, s2, …, s(i-1), s(I+1), …, sn)
L1 - How would define a players strictly best response / best reply to strategies s(-i) with notation? What would strange for a weakly best response?
The strategy s(i) that yields the player the greatest payoff, namely:
πi [si,s(-i)] > πi[si’, s(-i)] for any si ≠ si’
A weakly best response is a response at least as good as any other, so we would change the > sign for ≥.
L1 - When is an action strongly or strictly best response? What if it is not strictly best?
If no other actions are equally as good given the actions of the other players. Otherwise it is the weakly best response.
L1 - Mathematically, how would we define a strategy that is strictly dominant? In words?
πi [si,s(-i)] > πi[si’, s(-i)] for any si ≠ si’ and s-i
A dominant strategy would be a strictly best response with the extra caveat that it is for all s-i.
It is a strictly dominant strategy in for i if every other strategy of i is strictly dominated by it.
L1 - A dominant strategy equilibrium, in notation and analytically?
A strategy combination consisting of each players’ dominant strategy. It is the strategy that the players will choose as we assume they are rational players.
Mathematically, s^D ε S isa. strictly dominant strategy equilibrium if s^D ε Si is a strict dormant strategy for all i ε N.
Caveat, always refer to the solution as the actions the player takes, not the payoffs.
L1 - How, in words, could we descibe the composite parts of the function πi(si, s-i)?
This would be the payoff of the player i, using strategy si, given the strategies of all other players (s-i).
L1 - What does the failure of Pareto efficiency in the strictly dominant solution for the prisoners dilemma say about the strictly dominant solution concept? What does this lead to?
Not a lot. It cannot be seen as a failure as the players rationally make a choice given the information at their disposal.
The failure of pareto optimally implies that the players would benefit from modifying the environment in which they find themselves to create other enforcement mechanisms.
L1 - What alteration would we make to the payoff matrix with the introduction of an enforcement mechanism, such as a mafia?
Both players would have a further loss of ‘z’ if they chose to fink. The z amount will have to be large enough then the equilibrium outcome will be switched. and we can ensure a pareto optimality.
L1 - Claim: A rational player will never choose a dominated strategy.
Knowledge of the game implies that the player should recognise dominated strategies and rationality implies that these strategies will be avoided.
L1 - Why would the voting game lead to a weakly dependent equilibrium if P1 attains payoff 1 if A wins and player 2 attains a utility of 1 if B wins. Assume each attain a payoff of 0 otherwise. There are an odd number of other voters.
If they vote for their favourite candidate and majority of others also vote for it, they will received a payoff of 1, otherwise 0.
If they decide to vote for the alternative candidate, they may still receive a payoff of 1 if the majority of the other candidates vote for their preferred candidate. Therefore voting for the candidate weakly dominates not voting for them.
L1 - In evaluation, what can be said of the Weakly DSE? (4)
- It is similar to the DSE.
- When it exists, the solution concept generally guarantees uniqueness.
- It exists in a richer set of games than DSE.
- Has a similar pareto criterion outcome.
L2 - What does rationality imply (2)?
What can be said of rationality in relation to DSE?
- That rational players will never play a (strictly) dominated strategy.
- If a rational player has a dominant strategy equilibrium, he will play it.
Rationality is all that is required for DSE to hold.
L2 - Why does iterated elimination come into games?
The DS solution concept will often fail to exist. We therefore wish to develop a predictive theory that will apply in a wide variety of games. Given rationality, we are able to effectively reduce the number of viable options each player can take based on the dominated strafes, resulting in a “smaller” restricted game with fewer total strategies. It can also lead to strategies in the restricted game being dominated when they were not in the original game.
L2 - What is the process called in which we eliminate certain moves iteratively?
Iterated elimination of strictly dominated strategies.
IESDS.
L2 - What assumptions does IESDS require? (3)
- Rationality of all players;
- Common knowledge of rationality; and
- Common knowledge of game structure {N, S, π}
L2 - How would we denote the strategy profile that survives a IESDS? What would we call this profile?
See Brainscape Assistant 2.
An strategy profit that survives the process of IESDS is an iterated-elimination equilibrium (IEE)
L2 - How, in notation, would we express that a player I knows X?
κ(i)X
Where κ is Kappa and the brackets are used to identify the lower position of i with relation to κ.
L2 - What is the general process of IESDS? (4)
See Brainscape Assistant note 3.
L2 - Cournot Duopoly example for IESDS?
See Brainscape Assistant 4.
L2 - Relationship between IESDS and (S)DSE?
In a game {N, {Si}, {πi}}, if s* is a uniquely dominant strategy equilibrium, it uniquely shrives IESDS. e.g. the prisoners dilemma.
L2 - IEWDS for weakly dominated strategies notation?
See Brainscape Assistant 5. Notice that the S in IESDS is now a W, as we are referring to weakly dominated strategies.
L2 - What is a belief? How does the best response link into this?
A belief of player i is a possible profile of his opponents strategies, si ε S-i.
Given this belief, the player will be able to develop a best response given that belief.
L2 - A list of best responses?
A plan that maps beliefs into a choice of action, and this choice of action must be a best response to the beliefs.
L2 - What is the difficulty in basing best responses on beliefs?
It relies on the belief of what strategy P2 will take. We eliminate all other options and simply look at the best response in the column that for P1 to take. If P2 does not make chose the ‘belief’ P1 may end up worse off than using IESDS. Any response that is the best reply is rationalisable.
L2 - What is a rational response?
An action taken by a player under the belief that another rational player will choose a particular strategy. See Brainscape note 6.
L2 - Evaluate Rationalizability.
Similar to IESDS
- Existence: We can apply it to any game. It does not require strictly dominant or dominated strategies.
- Uniqueness: Anything could happen. (such as in the battle of sexes game)
- Pareto Efficiency: It may not be Pareto efficient.
L3 - What is a Nash equilibrium and its main assumptions?
A stable state in a game in which no participant can gain by a unilateral change in strategy.
- Each player is playing a best response to his beliefs. (this is a consequence of rationality)
- The beliefs of the players about their opponents are correct.
L3 - What are the alternative assumptions for a Nash Equilibrium (with notation as well)?
The player is choosing the best response to the equilibrium strategies of all other players.
See Brainscape Assistant 7
L3 - When would we be retired and to find and how could we find a Pure-Strategy Nash equilibrium?
This would be required if a strategy can not be attained via the dominant strategy, IESDS and rationalizability method.
Step 1: Find the best response of P1 given the actions available to P2.
Step 2: Find the best response of P2 given the possible actions of P1. If we have a strategy that is present in both steps, this will be the Pure-Strategy Nash Equilibrium.
L3 - Tragedy of the commons?
Refers to the conflict over scarce resources that restyle from the tension between individual selfish interests and the common good. See Brainscape Assistant 8.
L3 - Bertrand Duopoly?
A unique type of Nash equilibrium. Both firms set prices simultaneously. The result is a NE at the MC.
See Brainscape Assistant 9 or slide 14/26 in slide 3.
A simultaneous game where the strategic choice is on prices, rather than quantities (as would be the case in a Cournot game).
In this model, consumers will buy from the firm that offers the lowest price, so we can easily have the intuition that the Nash equilibrium is going to be the two firms setting the same price.
In the Cournot game, profits can be attained as a NE. In Bertrand, the two firms will either make negligible profits or if ε = 0, 0 profits are made.
L3 - Define in Notation a mixed strategy.
See Brainscape Assistant 10.
L3 - ΔSi meaning?
In mixed strategies, this is the set of distributions over set Si. A cumulative probability distribution.
L3 - σi meaning in mixed strategies?
It is the probability distribution over si.
σi = {σi(si1), σi(si2),…, σi(sim)}
σi(si1) is the probability that i plays si.
It assume a mixed strategy with the chance of each outcome being provided by a uniform distribution.
L3 - Given that σi is a probability distribution, what are the 2 facts to bear in mind when dealing with mixed strategies?
- σ s(i) ≥ 0 for all s(i) ε S(i)
- Σ(si ε Si) σi(si) = 1.
See slide 20/26 in L3 if confused .
L3 - What is the relationship between pure strategies and mixed strategies?
A mixed strategy for player I is just a probability distribution over his pure strategies. Every pure strategy is a mixed strategy with a degenerate distribution that
chooses a single pure strategy with probability 1.
L3 - How could we define a mixed strategy game in which the pure strategy sets are not finite? (more complicated notation)
Let Si be player i's pure-strategy set and assume it is an interval. A mixed strategy for player I is a cdf Fi: Si --> [0,1], where F(x) = Pr{si ≤ x}.
If Fi(.) is differentiable with density fi(s) then we say that si ε Si is in support of Fi(.) if fi(si)>0
See slide 21/26 in L3 for more information.L3 - Mixed strategies - Redefining Beliefs: pi(s-i)? With an example?
The belief that player i is given by a probability distribution pi ε ΔS-i over his opponents. Pi (s-i) is the probability that player I assigns to his opponents playing s-i ε S-i.
Example: Rock Paper Scissors.
The beliefs of player 1 are:
( p1(r), p1(p), p1(s) ).
With p1(r) ≥ 0, p1(p) ≥ 0, p1(s) ≥ 0 and p1(r) + p1(p) + p1(s) = 1.
L3 - What are the three applications of the Nash equilibrium?
- The Stag Hunt Game
- The Tragedy of the Commons (Hardin).
- Cournot Duopoly.
- Bertrand Duopoly
L3 - See picture taken on phone for F(i) and f(i)?
F(i) is the cumulative distribution function in a mixed strategy (with a the probabilities adding to 1). f(i) is the first differential, being the probability of attaining each amount.
Pr(s1 < s2) = 1 - F2(s1)
Pr(s1 > s2) = F2(s1)
L3 - All-pay action?
Define the game and explain the outcome.
One prize.
Two players simultaneously submit their bids (or could refuse to submit a bid).
The highest bid gets the prize.
Every player pays his bid.
If there is a tie, both players receive the prize with a portability of .5.
No equilibrium an be attained.
