Pure Strategy Flashcards
What is the specific definition of a NE in a simultaneous game?
Each player is simultaneously playing a mutual best response.
What assumption does Nash Equilibrium fundamentally rely on? What would make it a stronger equilibrium?
That players form correct beliefs about the other players’ actions/strategies.
A stronger solution would rely only on rationalisability - i.e actions/strategies that apply regardless of what the other person plays, so do not require perfect beliefs.
What is the specific definition of a best response?
An action/strategy for which
ui(ai,aj) >= ui(a’i, aj) for all possible actions of a’i that are a member of the action set A, holding the action of the other player(s) j fixed
What does Cournot model and what are the equilbria of the standard Cournot Model?
Models quantity-setting competition in oligopoly. Unique NE where quantity is an interior solution: q1=q2= a-c/3
[p(q) = a -q1-q2]
What is the difference between a best response correspondence and a function?
Correspondence - there may not always be a best response to an action
Function - always a defined best response for an action
[an input value (i.e action) can have only 1 outcome (does it have to be bijective?***)]
What does Bertrand model and what are the NE outcome(s)?
Price-setting competition in oligopoly. The only NE is where pi =pj =c. So profits = 0 in equilibrium.
Difference between Cournot and Bertrand?
- EQM Profits in Cournot are positive, in Bertrand they are equal to 0.
- Cournot models strategic substitutes, Bertrand strategic complements
- whilst Cournot has a unique best response for every case, Bertrand can produce either NO best response or MULTIPLE best responses
Similarity(ies) between Cournot and Bertrand model?
In monopoly, produces the same level of profit
What are strategic substitutes?
Strategies that move in the opposite directions to each other. e.g in Cournot, when one firm selected higher q, the other wanted to select lower q
What are strategic complements?
Strategies that move in the same direction as each other. e.g in Bertrand competition, each firm wanted to set prices close to the other, but slightly undercutting
Definition of a strictly dominant action
A ‘superior’ action that delivers strictly higher utility to a player, NO MATTER WHAT THE OTHER PLAYER PLAYS
Mathematical definition of strict dominance
ai’’ strictly dominates ai’ iff
ui(ai’’,aj) > ui(a’i, aj) for every list of actions j can take
Why wouldn’t a strictly dominated action appear in a NE?
The player playing this action would NOT be playing a best response. There is another action that delivers strictly higher utililty, for ALL actions of the other player.
Definition of weakly dominant action
The action is as least as good as the other action no matter what the other player plays, and is strictly better than the other action for some specific actions of the other player
Mathematical definition of weakly dominant actions
ai ‘’ weakly dominated ai’ iff
ui(ai’’,aj) >= ui(ai’, aj) for all aj
What is a strict NE?
Where no players’ equilibrium action can be weakly dominated by another action.
What is a non-strict NE ?
Where a player’s equilibrium can be weakly dominated by another action.
What is dominance solvability?
Where we have a unique solution following iterated deletion of strictly dominated actions
**How do we show that Cournot is dominance solvable?
What conditions do we need to guarantee the existence of a NE?
- Closed and bounded set of actions each player can take e.g [0,1]
- Continuous best response functions - i.e a unique best action that a player can take in response to another’s action
What lemma do we use to prove dominance solvability?
Suppose qj ∈ [q, q], then all qi > Bi(q) are strictly dominated by qi = Bi(q) and all qi < Bi(q)
are strictly dominated by Bi(q)
What is the essence of proving dominance solvability?
Start at the boundaries and work inwards,
What is the D(qj) function? (Dominance solvability)? What property does it have when we study the upper bound of a set?
The difference in firm i’s profits, when i selects a quantity q lower than qhat (where qhat is the best response to the bound)
D(qj ) = πi(ˆqi, qj ) − πi(qi, qj
D(qbar i) > 0 , as qhat is unique best response to the bound
What do we need to prove to show dominance solvability in Cournot?
That D(qj) > 0 for all qi < qˆi and all qj ∈ [q(Lbound), q (Ubound)]
How do we prove that D(qj) > 0 in the dominance solvability model?
Differentiate D(qj) wrt to qi get -qi(hat) + qi Decreasing function - therefore for qj < qbar, D(qj)>0 (as D(qbar) is already positive) qhat dominates all qi < qihat
In simple words explain Dominance solvability Lemma>
If j takes the lower bound value of a set, then i’s choice to set qi higher than their best response to j is strictly dominated by playing the best response
If j takes the upper bound value of a set, then i’s choice to set qi lower than their best response to j is strictly dominated by playing the best response.
What is the t’th stage best response to j playing qj = 0 in cournot (simultaneous)
B^t(0)= qm* [SUM of k=1 to t of: (-1/2)^k-1]
State the best response of firm i in a Bertrand model:
- Any pi > pj if pj < c
- Any pi ≥ c if pj = c
- Does not exist if pj ∈ (c, pM]
- pM if pj > pM
