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ECN306 - Game Theory > Game Theory > Flashcards

Game Theory Flashcards

1
Q

Define what makes a rational agent.

A

Agent is rational if they choose a€A that maximises payoff

a*€A is only chosen is v(a*) > v(a) for all a€A

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2
Q

What are the methods of evaluating solution concepts?

A

1) Existence - is there an equ?

2) Uniqueness - is there >1?

3) Invariance - is it likely to change

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3
Q

What are static games?

A

Set of players make independent, once and for all decisions after which the outcome is realised

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4
Q

What is complete information?

A

Common knowledge available among all players off all actions, outcomes, each players preferences over outcomes and impact of actions on outcomes.

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5
Q

What is a strategy profile?

A

Descriptions of pure stratefies for all players

eg. (s1, s4, s2) for players 1,2,3

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6
Q

What is pure strategy?

A

A deterministic plan of action

Set of possible pure strategies for player i is given by Si

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7
Q

What is a normal form game?

A

Includes:

  1. Finite set of players
  2. Collection of pure strategies - {S1, S2, .., Sn}
  3. Set of payoff functions - {v1, v2,.., vn}
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8
Q

What is pareto optimality?

A
  • A strategy profile s€S dominates s’€S if its payoffs are equal or higher for all players and at least one player’s payoff is higher.
  • vi(s) >= vi(s’) for all i €N
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9
Q

What is a strictly dominated strategy?

A

A strategy is strictly dominated by another if, for any combination of the other players’ strategies, another strategy will have a higher payoff.

vi(si, s-i) > v(s’i, s-i)

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10
Q

What is weak dominance?

A

A strategy is weakly dominated by another if the payoff is equal or less than another strategy given any strategy of the opponent.

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11
Q

What is IESDS and explain the steps.

A
  • IESDS = Iterated Elimination of Strictly Dominated Pure Strategies
  1. Identify strictly dominated strategies for all players
  2. remove these
  3. Treat whats left as a new game and repeat until there are no strictly dominated strategies
  4. Remaining strategic profile is the ‘iterated-elimination equilibrium’
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12
Q

How do you find the set of rationalizable strategies?

A
  1. Underline the best responses for each player in response to all others’ strategies
  2. If a row of P1 payoffs has no underlining, remove it
  3. If a column of P2 payoffs has no underlining, remove it
  4. Repeat to find BR
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13
Q

What did Keynes discover about IESDS in reality?

A
  • 1938 Beauty Contest
  • Two player choose number between 0 and 100
  • closest to 2/3 of average wins - prediction would be 0 with IESDS

Reasons why players did not choose 0:

  1. Cognitive bias - Nagel (1938)
  2. Chou (2007) found if hints are given, better performance
  3. Camerer (2003) - Iterations involved complex thinking, limited working memory.
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14
Q

What is a Nash Equilibrium?

A
  • Where each player is playing a best response to a belief and that belief is correct.
  • NO incentive to change strategy
  • vi(s*i, s*-i) >= v(s’i, s’-i) for all si€Si and all i€N
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15
Q

When do we get a unique Nash equilibrium?

A

When we have any of:

  1. Strict dominant equilibrium
  2. Sole survivor of IESDS
  3. Unique rationalizable strategy
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16
Q

What are the requirements for a Nash Equilibirum?

A
  1. Each player is playing a BR to their belief
  2. Beliefs about opponents are correct
17
Q

What are some properties of Nash Equilibria?

A
  1. Stability - no incentive to deviate
  2. Self fulfilling - if there are multiple NE & players agree on one, they wont deviate
18
Q

What are some limitations of Nash Equilibria?

A
  1. No guarantee of existence
  2. Insecurity about others actions
  3. Can be multiple NE and would need coordination to achieve one
19
Q

What can happen when there are multiple Nash Equilibria (multiplicity)?

A
  • If coordination is allowed we can get agreements even if they arent enforced
  • Repition may lead to coordination
  • A pareto dominant equilibrium may arise
20
Q

What is the concept of the maxmin strategy?

A

Reduces uncertainty - Player is finding the best worst case scenario, so opponents actions are less likely to affect them negatively

21
Q

How do you find a maxmin strategy?

A
  1. Find the minimum possible payoff for each strategy
  2. Play the strategy with the highest value of these minimums
22
Q

What is a mixed strategy?

A
  • Where a player plays a probability distribution over their strategies
  • mixed strategy given by σi={ σi(si1), σi(si2), …, σi(sim) }
  • eg for Rock paper scissors {R, P, S}, mixed strategy could look like σ = { 1/3, 2/3, 0}
23
Q

What are the properties of a probability distribution in mixed strategies?

A
  1. all probabilities are above or equal to 0
  2. All probabilities sum to 1
24
Q

What is an expected payoff in mixed strategies?

A
  • Calculated expected payoff from a given strategy given the beliefs of your opponents strategies
  • vi(si,σ-i) = Σ payoff x probability = Σvi(si,s-i)σ-i(s-i)
25
Q

What is the mixed strategy equilibrium?

A
  • Nash equilibrium if for each player their mixed strategy is the best response to their opponents mixed straegy
  • In NE players will be indiff between their pure and mixed strategies
  • P1 chooses probabilities to make P2 indifferent
  • [{P, (1 - p) } , {q, (1 - q) }]
26
Q

How do you find the mixed straegy equilibrium in a 3x3 game?

A
  • A pure strategy will be dominated by a mixed stragegy such as {0, 0.5, 0.5}
  • Eliminate this dominated strategy
  • repeat until at a 2x2 matrix
  • then solve normally using P and q
27
Q

Why would you use a mixed strategy equilibrium?

A
  1. Von Neumann, Morgenstern (1944) To keep the opponent guesssing/ hide your intentions
  2. Gibbons (1992) because of uncertainty about the other player
28
Q

How does mixed strategy equilibriumwork in reality?

A
29
Q

How do you find the sequentially rational Nash equilibrium in a dynamic game?

A
  1. Draw the matrix and the tree
  2. Make sure the matrix has the correct pure strategies {oo,of,fo,ff}
  3. Find the nash equilibria in the matrix
  4. Use backwards induction to make sure that the pure strategy is a best response both on and off the equilibrium path
30
Q

How do you find a subgame perfect nash equilibrium?

A
  1. Draw the matrix and the tree
  2. Where there is an information set with more than one node make it into a subgame
  3. Find the nash equilibrium in the matrix
  4. Use backwards induction to find the SPNE
  5. SPNE implies there is a NE in each subgame both on and off the equilibrium path
31
Q

How do you calculate the total payoff for an individual in a finitely repeated game?

A
32
Q

How can you generate incentives to deviate from NE in repeated games?

A
  • By punishing the individual from deviating from a pareto superior strategy
  • Make the expected payoffs from deviating lower than sticking to the strategy
  • vi* > vi’
  • There will be a critical vallue of d for this
  • Also needs two distinct NE in the final stage
33
Q

How do you calculate the present value of total payoffs for an infintely repeated game?

A
34
Q

What are Grim - Trigger strategies?

A
  • Player starts with the good strategy
  • Play this strategy until deviation is detected
  • Then plays the bad strategy forever
  • for there to be no incentives to deviate vi* > vi’
    *
35
Q

What is a Bayesian game?

A
  • A game with incomplete information
  • Where a player does know who the other players are, doesnt know their actions, and/or doesnt know how their actions translate into payoffs
  • Nature chooses a profile of types, players know their own
  • Common prior assumption - players know precse distribution of types
36
Q

What is Harsanyi’s Method?

A
  • Transform a game of incomplete information into a game of imperfect information
  • Produce a meta player who allows average/expected payoffs to be calculated
  • Does this by assigning probabilities to types
  • The type of the player is private info so information sets may be needed
  • Expected payoffs are calculated using probability theorum, then payoffs are maximised using NE

ECN306 - Game Theory (1 decks)

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