Best Response Functions

Purpose

• we are best interested in determining the best actions for a player given the actions of the other players

• for any player i, the best response function associates a set of actins by player i to any list of the other players’ actions

Best Response Function

• B_i(a_{-i}) = {a_i in A_i u_i(a_i, a_{-i}) >= u_i(a’_i, a_{-i}) for all ‘_i in A_i}

  • where A_i is the set of possible actions for player i

• Note

  • every member of the set B_i(a_{-i}) is a best response of player i to a_{-i}

  • if other players choose a_{-i} then player i can do no better than to choose a member of B_i(a_{-i})

Proposition 36.1

the action profile a* is a NE if a strategic game with ordinal preferences iff every player’s action is a best response to the other players’ actions: a*_i is in B_i(a*_{-i}) for every player i

Special Case: Single Best Response

• if each player i has a single best response to each list a_{-i} we can write (36.2) as equations:

  • a*_i = b_i(a*_{-i}) for every player i

    • where b_i(a*_{-i}) is the single member in B_i(a*_{-i})

  • (36.3) is a set of n equations in n unknowns (where n is the number of players)

Equation 36.2

a*_i is in B_i(a*_{-i}) for every player i

Using BRF to Find NE

2 ways

1. finding the best response function for each player

2. finding the action profiles that satisfy (36.2) which reduces to (36.3) if all playesr have single best responses