Games Markets & Information Flashcards
1
Q
L1 - What are the 4 features of a normal form game, with notation?
A
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
2
Q
L1 - Normal Form Game Feature definition: Players?
A
Are individuals who make decisions.
3
Q
L1 - Normal Form game Feature definition: Actions?
A
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.
4
Q
L1 - Normal Form game Feature definition: Payoffs?
A
By payoff, we mean the utility player i receives after all players have picked their strategies and the game has been played out.
5
Q
L1 - Normal Form game Feature definition: Information?
A
Specifies what player I knows at different points in time. Usually modelled as information sets.
6
Q
L1 - Strategy in the terms of games?
A
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.
7
Q
L1 - For a Set of s* = s1, s2, …, sn, what does each s represent? What is the definition of the s equilibrium?
A
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.
8
Q
L1 - What are the three sets we use in a normal form game and their notation?
A
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).
9
Q
L1 - How would you represent the payoffs for each player in the prisoners dilemma using the symbols in the slides?
A
[π1(c,c), π2(c,c)]
[π1(s,c), π2(c,s)]
[π1(c,s), π2(s,c)]
[π1(s,s), π2(s,s)]
10
Q
L1 - How would we denote (symbols) of a normal form representation game?
A
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}
11
Q
What does the payoff matrix represent?
A
The utilities of each player given the strategies taken by each player.
12
Q
L1 - What is the table called that represents the utilities of N players given their strategies?
A
The payoff matrix.
13
Q
L1 - What is the difference between Si and si?
A
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.
14
Q
L1 - S-i?
A
The list of strategy sets of all players excluding player i.
15
Q
L1 - What is the payoff notation for players i’s best response (s*i) given all the other platers responses?
A
πi (si, s-i) Is greater than or equal to πi(s’i, s-i) for an S’i not equal to si.
16
Q
L1 - The difference between strongly (strictly) best response and weakly best response?
A
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.
17
Q
L1 - When is strategy s*I a dominant strategy? Define and denote.
A
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.
18
Q
L1 - What is a dominant strategy equilibrium?
A
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
19
Q
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?
A
BR(S) = C
20
Q
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.
A
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.
21
Q
L1 - What are strictly dominated strategies and what is the assumption?
A
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.
22
Q
L1 - Why might the Dominant Strategy Equilibrium fail?
A
Some games do not have a dominant strategy for players, and thus we do not attain a dominant strategy equilibrium.
23
Q
L1 - How can we link in the Pareto Criterion into the solution concept (strictly dominant equilibrium) and what would the implication be?
A
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.
24
Q
L1 - Enforcement Mechanism?
A
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.
25
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.
26
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.
27
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.
28
L1 - What do we mean by 'pure' in pure strategies?
Players choose deterministic action, namely that players choose a certain plan of action.
29
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.
30
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.
31
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.
32
L1 - When would a payoff matrix not be able to be used?
When the game is not finite.
33
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)
34
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 ≥.
35
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.
36
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.
37
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.
38
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).
39
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.
40
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.
41
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.
42
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.
43
L1 - In evaluation, what can be said of the Weakly DSE? (4)
1. It is similar to the DSE.
2. When it exists, the solution concept generally guarantees uniqueness.
3. It exists in a richer set of games than DSE.
4. Has a similar pareto criterion outcome.
44
L2 - What does rationality imply (2)?
| What can be said of rationality in relation to DSE?
1. That rational players will never play a (strictly) dominated strategy.
2. If a rational player has a dominant strategy equilibrium, he will play it.
Rationality is all that is required for DSE to hold.
45
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.
46
L2 - What is the process called in which we eliminate certain moves iteratively?
Iterated elimination of strictly dominated strategies.
| IESDS.
47
L2 - What assumptions does IESDS require? (3)
1. Rationality of all players;
2. Common knowledge of rationality; and
3. Common knowledge of game structure {N, S, π}
48
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)
49
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.
50
L2 - What is the general process of IESDS? (4)
See Brainscape Assistant note 3.
51
L2 - Cournot Duopoly example for IESDS?
See Brainscape Assistant 4.
52
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.
53
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.
54
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.
55
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.
56
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.
57
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.
58
L2 - Evaluate Rationalizability.
Similar to IESDS
1. Existence: We can apply it to any game. It does not require strictly dominant or dominated strategies.
2. Uniqueness: Anything could happen. (such as in the battle of sexes game)
3. Pareto Efficiency: It may not be Pareto efficient.
59
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.
1. Each player is playing a best response to his beliefs. (this is a consequence of rationality)
2. The beliefs of the players about their opponents are correct.
60
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
61
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.
62
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.
63
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.
64
L3 - Define in Notation a mixed strategy.
See Brainscape Assistant 10.
65
L3 - ΔSi meaning?
In mixed strategies, this is the set of distributions over set Si. A cumulative probability distribution.
66
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.
67
L3 - Given that σi is a probability distribution, what are the 2 facts to bear in mind when dealing with mixed strategies?
1. σ s(i) ≥ 0 for all s(i) ε S(i)
2. Σ(si ε Si) σi(si) = 1.
See slide 20/26 in L3 if confused .
68
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.
69
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.
```
70
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.
71
L3 - What are the three applications of the Nash equilibrium?
1. The Stag Hunt Game
2. The Tragedy of the Commons (Hardin).
3. Cournot Duopoly.
4. Bertrand Duopoly
72
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)
73
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.
74
L3 -ΔSi meaning for Heads and Tails
ΔSi = { (σi(H), σi(T) : σi(H) > 0 , σi(T) > 0, σi(H) + σi(T) = 1 }
A mixed strategy.
75
L3 - Definition: Define the NE for a more general set up:
| σ* = (σ*1, σ*2, ..., σ*n).
The mixed-strategy pro
le
σ* = (σ*1, σ*2, ..., σ*n)
is a Nash equilibrium if for each player σ*i is a best response to σ*-i
That is, for all i ε N:
πi( σ*i, σ*-i) ≥ πi (σ*I, σ*-i) for any σi ε ΔSi
76
L3 - For πi( σ*i, σ*-i), how can we interpret σ*-I?
The belief f later I about his opponents, pi choices of the pure strategies. The profile of mixed strategies, σ*-i, captures the beliefs over all of the pure strategies that i's opponents can play.
77
L3 - What are the two assumptions we hold when concluding anything about the Mixed-Strategy Nash equilibrium?
Rationality requires that a player plays a best response given his beliefs.
A Nash equilibrium requires that these beliefs are correct.
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L4 - What do we allow players to do with mixed strategies?
Randomize between different actions.
85
L4 - What is the matching pennies game and what would the payoff matrix be?
Two players put a coin on the table. If the coins are matching, P1 gets both, otherwise, P2 does. See Brainscape Assistant 6 for the payoff matrix.
86
L4 - What can be said about the matching pennies game in relation to pure strategies? What about Rock Paper Scissors?
There is no (pure-strategy) Nash equilibrium, however, a Nash equilibrium exists if we allow players to choose random actions. With rock, paper scissors there is no Nash equilibrium.
87
L4 - How would we indicate a player, i's, finite set of pure strategies and player i's mixed strategy? Use an example of rock, paper scissors.
Pure:
Si = {si1, si2, ..., sim}
Si = {Rock, Paper, Scissors}
Mixed strategy:
σi = {σi(si1), σi(si2), ..., σi(sim)}
𝛔1 = {G1(R), G2(P), G3(S)}
Being the probability distribution over Si, σi(si) is the probability that I plays si.
88
L4 - Do the pure strategy sets need to be finite in Bertrand Duopoly's, Cournot Duopoly's or other such games?
No, in most cases, a mixed strategy will be given by a cumulative distribution function.
89
L4 - How would we indicate the cumulative distribution function for players mixed strategy?
More simple notation.
Fi : Si --> [0,1] Where F(x) = Pr {si ≤ x}.
The probability density function is indicated by fi(si), which, if Fi is differentiable, we say si ε Si in support of Fi (x) if fi(si) > 0.
90
L4 - With the matching Pennies game, assume P2 picks σ2(H) : 1/3 and σ2(T) = 2/3. What would the EP be with a σ1(H): 1/3 and σ1(T): 2/3 vs σ1(H): 2/3 and σ1(T): 1/3.
What does the above case describe?
σ1(H): 1/3 and σ1(T): 2/3
If he plays H, he will get a payoff of 1 with a probability of 1/3, and -1 with a probability of 2/3.
σ1(H): 2/3 and σ1(T): 1/3.
If he plays H, he will get a payoff of 1 with a probability of 2/3, and -1 with a probability of 1/3.
This case describes lotteries.
91
L4 - Definition: When is a pure strategy played in a mixed strategy?
A pure strategy si ε Si is in the support of σi if σi(si) > 0,
or more simply, if the mixed strategy of i is played with positive probability.
92
L4 - When would a player be willing to randomise between two strategies in for a mixed strategy?
When two or more si are in the support of σ*i.
If the Nash equilibrium pro
file σ*, with the support of i's mixed strategy contains more than one pure strategy, i.e. si and s'i are both in the support of σ*i.
Since σ*i is a best response against σ*-I player i cannot do better than to randomize between more than one of his pure strategies, si and s'i .
93
L4 - Explain the 'reporting a crime' game. What can we say about the
Players: n
Actions: Call Police, Don't Call Police.
Preferences: 0 if no-one calls, v if one person calls, v - c if the chosen individual calls.
94
L4 - 'reporting a crime' game - What can we say about the pure asymmetric and symmetric NE? What is the difference?
What about a mixed strategy?
Asymmetric (Players can play different strategies) - There are n pure-strategy NE, which results in one person calling in each NE.
Symmetric (all players play the same strategy) - No game has a symmetric NE.
We have a symmetric mixed strategy equilibrium if each person calls with a positive probability less than 1.
95
L4 -'reporting a crime' game - Show the equilibrium condition if 'v - c' is a person's payoff to calling, 0 if no-one calls, and v if someone else calls.
A person's payoff to calling must be equal to their payoff to not calling.
v: payoff from someone calling
c: cost to call.
v - c = 0 * Pr{no-one calls} + v * Pr{at least one other person calls}
v-c = v(1-Pr{No-one else calls })
c/v = Pr {No-one else calls}
See Brainscape Note 9 for the remainder.
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L4 - When is NE the minimal necessary condition for reasonable predictions?
In mixed strategies in which players use independent randomisation to arrive at their mutual actions.
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L4 - How could two players do better than randomising between two options when there exist two NE?
By using a public or private signalling mechanism, such as a toss of a coin, in which they must both adhere to the given strategy on flipping. The overall payoff could then be improved.
Note: This is a public signal.
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L4 - What would an example of a public signal be?
A flip of a coins, which leads players to undertake a given strategy.
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L4 - What can a private randomisation device allow?
It make make payoffs available outside the cones hull of Nash equilibria attainable.
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L5 - Extensive Form Games of Perfect Information - Common features? (3)
1. Moves occur in sequence;
2. All previous moves are observed before a move is chosen;
3. Payoffs and structure of the game are common knowledge.
114
L5 - Extensive Form Games of Perfect Information - What are the four components of extensive games?
1. Set of players, N.
2. Terminal Histories (A set of sequences).
3. Player function
4. Preferences of the players. (Player 1's preferences would be U1(Action) = payoff, U2(Action2) = Payoff 2 etc).
(2 and 3 are new concepts)
115
L5 - Extensive Form Games: Player Function?
Specifies at any stage of the game who is going to play.
It assigns the player to every sequence that is a proper subhistory of some terminal history.
e.g. The entry game:
A challenger would be assigned to the start of a game and the incumbent would be limited to the point in which the challenger enters the game.
P(⊘) = Challenger and P(In) = Incumbent
116
L5 - Extensive Form Games: Terminal Histories?
Define
Property
Example: Entry Game
A set of sequences (terminal histories) - A list of played actions from the beginning of the game to the end.
Property: No sequence is a proper subhistory of any other sequence.
e.g. In the entry game
(In,accomodate) (In,Fight) (Out)
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L5 - Extensive Form Games: ⊘?
A subhistory of a finite sequence consisting of no actions (representing the start of the game).
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L5 - Extensive Form Games: Proper Sub History? *****
A sub history not equal to the entire sequence.
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L5 - Extensive Form Games: History? **
A sequence of actions that is a a subhistory of some terminal history.
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L5 - Γ?
Perfect Information. Often referring to an extensive form game with perfect information.
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L5 - Extensive Form Games: The Entry Game - What would the sub history of (in, accommodate) be?
The Empty History ⊘ and the sequences (In) and (In, Accommodate)
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L5 - Extensive Form Games: The Entry Game - What are the Proper Subhistories?
The Empty History ⊘ and the sequence (In)
| The Proper Subhistory is contained within the history.
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L5 - Decision Tree: Decision Node?
On the node will be the name of the player who acts.
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L5 - Decision Tree: Branches following a decision node?
The actions of the player given by the decision node.
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L5 - EFG - A(h) = {a : h, a}
| What does each part of the above represent?
A set of Actions of a player.
A(h): The A (actions) of player , who moves after h, where h is an action taken.
{a : the actions available to the player who plays after h.
(h, a): the sequence, being a history of all actions for the player moving after h.
e.g. The Entry Game
The player who moves at the start of the game has actions:
A(⊘) = {In, Out}
The actions of the player who moves after the history In, the incumbent, is A(In) = {Accommodate, Fight}
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L5 - EFG - Strategies?
A player's strategy specifies the action the player chooses for every history after which it is its turn to move.
It is a function that assigns to each history, h, after which it is player i's turn to move, an action.
e.g.
P(h) = I where p is the player function, an action A(h), being the set of actions available after h.
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L5 - EFG - Strategies - What is the implicit assumption behind strategies?
Plan of Action.
A player's strategy provides sufficient information to determines its plan of action: the actions it intends to take, whatever the other players do. In particular, if a player appoints an agent to play the fame for it, and tells the agent its strategy, there the agent has enough information carry out its wishes, whatever actions the other players take.
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L5 - EFG - What is the relevance of Normal Form Games?
All games can be translated into normal form games to then find the NE.
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L5 - How can we represent extensive form games with perfect information?
1. Translate them into normal form games (payoff matrix)
| 2. As a decision tree.
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Extensive Form Games - Nash Equilibrium Definition? **
SEE Definition on L5 13/16
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L5 - What would {O, F|O, F|F} be an example of?
| What would the terminal history be?
A strategy profile.
The terminal history of this strategy profile would be {O, F}
It is represented in a pure strategy.
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L5 - Definition: When would a strategy profile s* in an extensive game with perfect information be a Nash equilibrium?
If, for every player, i, and every strategy, si', of player I, the terminal history O(s*) generated by s* is at least as good according to player i's preferences as the terminal history O(s'i, s*-I generated by the strategy profile (s'i, s*-i) in which player I chooses s'i, while every other player j chooses sj*.
πi((O(s*)) ≥ πi(O(s'i, s*-i)) for every strategy s'i of player i.
πi is a payoff function that represents player i's preferences and O is the outcome function of the game.
133
L5 - In Extensive Form Games, when do we come across the term credible threat?
When we translate the games into normal form and look for equilibriums, there are certain strategies that lead to a NE when we look at a specific action of the player. We basically remove the chance of one part the strategy from occurring as we assume P1 picks the other action.
The player to move second may in fact not credibly be able to convince P1 that they will play a given strategy if it resulted in P2 being worse off. See slide 7/27 of L5 for more information.
134
L5 - Extensive Games - What would the subgame following a nonterminal history h be?
The part o the game that remains after h has occurred. It can be analysed on its own.
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L5 - SEE Subgame definition in L5 on page 8/27 and take notes.
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L5 - See above note
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L5 - See above note
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L5 - What problem does the subgame perfect equilibrium solve?
It gets rid of non credible threats.
139
L6 - What symbol do we use when describing a sub game equilibrium strategy profile?
σ
140
L6 - Definition: When is the strategy profile σ* = (σ*1, ...,σ*n) of an extensive form game Γ called a subgame perfect equilibrium?
If it prescribes a NE in each of its proper subgames. This implies that in no subgame can any player do better by choosing a strategy different from σ*I, given that every other player j follow σ*j.
141
L6 - How many subgames are there in the sequential battle of the sexes game?
3:
| O|O, F|F and the entire game.
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L6 - What can be said about a subgame perfect NE in relation to a NE for the entire game?
Any subgame perfect NE must also be a NE for the entire game. This is what rules out any strategies that lead to non-credible threats.
143
L6 - What are the 3 things we must do to find a Subgame Perfect equilibrium?
1. Identify all its proper subgames.
2. Find NE in each proper subgame (using its normal form representation).
3. Among the NE, find the profile of strategies that induce a NE in all proper subgames.
144
L6 - Explain the process for Backward Induction to find the sub game perfect NE. (4)
1. Define the length of a subgame to be the length of the longest history in the
subgame.
2. Replace all subgames of length 1 by their Nash Equilibrium outcomes. This creates a new set of subgames of length 1 in a shortened game.
3. We again replace all subgames of length 1 by their Nash Equilibrium
outcomes. This creates the new set of subgames of length 1 in a shortened game.
4. We continue repeating this reduction until the game only constitutes of a
starting node and a set of actions leading to terminal nodes. The NEq'm
outcome of this reduced game is the backward induction outcome of the
original game.
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L6 - What is a benefit and drawback of the subgame perfect equilibrium?
Benefit: Backwards induction always leads to a strategy profile which is a subgame perfect equilibrium.
Drawback: It does not work in games with information sets which contain more than one node. In complicated games the process may be very difficult, so we can use a simpler method.
146
L6 - ** NEED TO CREATE A BACKWARD INDUCTION GAME MYSELF USING THE METHOD DESCRIBED IN L5 SLIDE 15/37
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147
L6 - What is a singleton?
***
148
L6 - Sequential Bargaining Game - Explain the game.
2 Player
Single (divisible) object - $1 (s)
Alternating Offers.
Player 1 makes an offer on how to split the $1, P2 accepts of rejects. If P2 rejects, they can then make an offer back.
This continuities until the $1 is split.
Any division of the object x* ε [0,1], (s1, s2) = (x*, 1 - x*) can be supported by a NE.
Each offer take one period and is irrelevant once rejected.
Player are impatient (discount factor 0 < 𝛿 < 1)
149
L6 - Explain the Game of Sequential Bargaining when T < ∞ period bargaining and T is an off number. *
See L6 slide 20/37 * No NOTE TAKEN. GO OVER. CANNOT LOCATE IN BOOK.
Note that
150
L6 - TIOLI?
Slide 19/37 *
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L6 - Slides 21,22,23
*
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L6 - Slides 21,22,23
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153
L6 - Slides 21,22,23
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154
L6 - Slides 21,22,23
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155
L6 - Slides 21,22,23
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156
L6 - Slides 21,22,23
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L6 - Slides 21,22,23
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L6 - Slides 21,22,23
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159
L6 - With Sequential Bargaining, what can we say about the infinite horizon game?
What is the equilibrium for even periods?
What is the equilibrium for odd periods?
There is no last period, so we can't apply backward induction.
However, in case of infinite horizon, the subgame that starts at period 3 after rejecting the offer from the first two rounds is identical to the entire game, Therefore, the offer that was optimal at period 1 is also optimal at period 3, and the shares of both players in period three and period 1 are the same. We attain the share:
(s*1, 1-s*1), = (1 - 𝛿 (1 - 𝛿s*, 𝛿 (1 - 𝛿 s*))
Therefore,
(s*, 1 - s*) = (1 / 1 + 𝛿) , 𝛿 / 1 + 𝛿 )
Odd Period: P1 offers 1 / 1 + 𝛿, leaving 𝛿 / 1 + 𝛿 to P2. P2 accepts this offer.
160
L6 - What is the definition of imperfect information?
An information set I(h), consists of histories, h, which are indistinguishable to the player who moves at I(h).
161
L6 - What is an info set? What is the result on actions?
The set of nodes between which the moving player cannot distinguish at any given time.
(imperfect information).
Since the player cannot distinguish between any nodes in an info set, a strategy can only prescribe one action per info set.
162
L6 - What situations do games of complete but imperfect information allow us to model? (2)
1. Players do not observe the moves of some previous agents.
| 2. Moves by Nature (exogenous uncertainty)
163
L6 - L5 - Extensive Form Games of Imperfect Information - Common features? (4)
Two features different to the game with perfect information.
We had:
1. Moves occur in sequence;
3. Payoffs and structure of the game are common knowledge.
We add:
1. The knowledge that players have when they can move
2. Probability distributions over exogenous events.
164
L6 - Subgame Perfect equilibria - How can this be attained when we have subgames?
We eliminate all strategies which are not equilibrium strategies in all subgames. So, we get rid of empty threats.
165
L6 - For Slides 36 - 37
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166
L6 - For Slides 36 - 37
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167
L6 - For Slides 36 - 37
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168
L6 - Extra
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169
L6 - Extra
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170
L6 - Extra
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171
L6 - Extra
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172
L6 - Extra
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173
L7 - When we have a game of incomplete information, in which a player does not now the cost of another player in building a plant. How does Harsanyi propose to model such a situation? (4)
1. Transform the static strategic form game of incomplete information into an extensive form game.
2. Add an addition Player N (nature) with at least once chance move at the beginning of the extensive form game.
3. This player N selects the unknown payoff relevant parameter according to a public probability distribution.
4. This move by N is observable to the privately informed player, but not to anyone else.
174
L7 - Harsanyi Model - What assumptions do we hold? (4)
| What is the 4th assumption commonly known as?
1. Players maximise their payoffs even their beliefs about the situation they are in.
2. We assume that layers know their own preferences but do not know the preferences of or types of their opponents.
3. Each player knows the precise way in which nature chooses these preferences.
4. Each players knows the probability distribution over types, and this itself is common knowledge among the players of the game - commonly known as the COMMON PRIOR ASSUMPTION.
175
L7 - Harsanyi Model - Once we have transformed the static strategic form game of incomplete information into an extensive form game and completed his approach, how would we find the solution to such a problem?
*
1. Draw the Tree digram with N at the tope and then the work down in order of the sequential turns.
2. Define q as the probability that the action of P2 occurs, (1 - q) being the probability it doesn't occur.
3. Use the payoffs of player 1 based on the assumption that another player takes an action or not to create an equation and solve.
4. Represent that best course of action for player 1 given the probability established in 3.
Define a new variable, say r to be Pr (action of P1 | Cost of action for player 1) and z = Pr action of player 1 | higher cost of action by player 1) - NOTE - one of these actions may be strictly dominated so can be excluded.
*
Complete once L7 Sldie 8/24 has been resolved.
176
L7 - How does the Harsanyi Model transfer games of incomplete incomplete information? What does it rely on?
It transforms games of incomplete information into games of imperfect information.
The approach is based on the very strong common prior assumption (the distribution of types is common knowledge).
177
L7 - What is the roadmap for games of incomplete information? (3)
1. We modelled this situation as one in which players have uncertainty about the
preferences of other players.
2. We assumed that players share the same beliefs about this uncertainty
(common prior assumption)
3. For any behaviour of his opponents, every player can calculate his expected
payoffs from any action.
178
L7 - In simple terms, what does the Harsanyi approach do and what is its assumption?
It transforms games of incomplete information into games of imperfect information.
It relies on the very strong common prior assumption (the distribution of types is common knowledge.
179
L7 - What sequence can be used to describe a static Bayesian game? (4)
Nature chooses a profile of types (θ1, θ2 ,..., θn)
2. Each player I learns his own type (high cost or low cost in the slides example) , θi, which is his private information, and then used his prior Φi to form posterior beliefs over the other types of players.
3. Players simultaneously choose actions ai ε Ai for all I ε N.
4. given the players' choices, a = (a1, a2, .., an), the payoffs πi(a,θi) are realised for each player.
180
L7 - Bayesian Games of incomplete information - How are posterior beliefs derived Φi (Φ-i|Φi)? (2)
1. Before Nature chooses the actual type of each player, every player knows the probability distribution that Nature used to chooses the types for all the players (prior).
2. After Nature has chosen a type for each player, players independently and privately learn their types. For every player, his type, may provide some information about how the other players' types may have been chosen, leading to the potential for a player to derive new beliefs about the other players once he learns his type.
181
L7 - Common prior and posteriors - What is the Bayes rule?
Use an example using the following payoff matrix:
1. 2.
1. 1/6. 1/3
2. 1/3. 1/6
The conditional on event S being true, the conditional probability that event H is true given by:
Pr (H | S) =
Φ (SnH) / Φ(S)
Example:
Φ1(Φ2 = 2 | Φ1=1) =
Pr (Φ2 = 2 n Φ1=1) / Pr( Φ1=1) = 1/3 / (1/3 + 1/6) = 2/3
182
L7 - Bayesian Form Games - How do actions and strategies differ?
They are not the same.
An action is the set of choose a privately informed player has (after learning his type) in the original game of incomplete information.
A strategy is a rule for player i to choose an action as function of his type.
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L7 - Bayesian Form Games - What is a mixed strategy in relation to a pure strategy
A mixed strategy is a probability distribution over player's pure strategies.
