Game Theory Flashcards
Define what makes a rational agent.
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
What are the methods of evaluating solution concepts?
1) Existence - is there an equ?
2) Uniqueness - is there >1?
3) Invariance - is it likely to change
What are static games?
Set of players make independent, once and for all decisions after which the outcome is realised
What is complete information?
Common knowledge available among all players off all actions, outcomes, each players preferences over outcomes and impact of actions on outcomes.
What is a strategy profile?
Descriptions of pure stratefies for all players
eg. (s1, s4, s2) for players 1,2,3
What is pure strategy?
A deterministic plan of action
Set of possible pure strategies for player i is given by Si
What is a normal form game?
Includes:
- Finite set of players
- Collection of pure strategies - {S1, S2, .., Sn}
- Set of payoff functions - {v1, v2,.., vn}
What is pareto optimality?
- 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
What is a strictly dominated strategy?
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)
What is weak dominance?
A strategy is weakly dominated by another if the payoff is equal or less than another strategy given any strategy of the opponent.
What is IESDS and explain the steps.
- IESDS = Iterated Elimination of Strictly Dominated Pure Strategies
- Identify strictly dominated strategies for all players
- remove these
- Treat whats left as a new game and repeat until there are no strictly dominated strategies
- Remaining strategic profile is the ‘iterated-elimination equilibrium’
How do you find the set of rationalizable strategies?
- Underline the best responses for each player in response to all others’ strategies
- If a row of P1 payoffs has no underlining, remove it
- If a column of P2 payoffs has no underlining, remove it
- Repeat to find BR
What did Keynes discover about IESDS in reality?
- 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:
- Cognitive bias - Nagel (1938)
- Chou (2007) found if hints are given, better performance
- Camerer (2003) - Iterations involved complex thinking, limited working memory.
What is a Nash Equilibrium?
- 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
When do we get a unique Nash equilibrium?
When we have any of:
- Strict dominant equilibrium
- Sole survivor of IESDS
- Unique rationalizable strategy
What are the requirements for a Nash Equilibirum?
- Each player is playing a BR to their belief
- Beliefs about opponents are correct
What are some properties of Nash Equilibria?
- Stability - no incentive to deviate
- Self fulfilling - if there are multiple NE & players agree on one, they wont deviate
What are some limitations of Nash Equilibria?
- No guarantee of existence
- Insecurity about others actions
- Can be multiple NE and would need coordination to achieve one
What can happen when there are multiple Nash Equilibria (multiplicity)?
- If coordination is allowed we can get agreements even if they arent enforced
- Repition may lead to coordination
- A pareto dominant equilibrium may arise
What is the concept of the maxmin strategy?
Reduces uncertainty - Player is finding the best worst case scenario, so opponents actions are less likely to affect them negatively
How do you find a maxmin strategy?
- Find the minimum possible payoff for each strategy
- Play the strategy with the highest value of these minimums
What is a mixed strategy?
- 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}
What are the properties of a probability distribution in mixed strategies?
- all probabilities are above or equal to 0
- All probabilities sum to 1
What is an expected payoff in mixed strategies?
- 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)
What is the mixed strategy equilibrium?
- 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) }]
How do you find the mixed straegy equilibrium in a 3x3 game?
- 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
Why would you use a mixed strategy equilibrium?
- Von Neumann, Morgenstern (1944) To keep the opponent guesssing/ hide your intentions
- Gibbons (1992) because of uncertainty about the other player
How does mixed strategy equilibriumwork in reality?
How do you find the sequentially rational Nash equilibrium in a dynamic game?
- Draw the matrix and the tree
- Make sure the matrix has the correct pure strategies {oo,of,fo,ff}
- Find the nash equilibria in the matrix
- Use backwards induction to make sure that the pure strategy is a best response both on and off the equilibrium path
How do you find a subgame perfect nash equilibrium?
- Draw the matrix and the tree
- Where there is an information set with more than one node make it into a subgame
- Find the nash equilibrium in the matrix
- Use backwards induction to find the SPNE
- SPNE implies there is a NE in each subgame both on and off the equilibrium path
How do you calculate the total payoff for an individual in a finitely repeated game?
How can you generate incentives to deviate from NE in repeated games?
- 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
How do you calculate the present value of total payoffs for an infintely repeated game?
What are Grim - Trigger strategies?
- 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’
*
What is a Bayesian game?
- 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
What is Harsanyi’s Method?
- 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
