# Best Response Functions for Cournot

For a Cournot Duopoly game with Linear Demand functions, we need to find the optimal quantity of each firm such that profit is maximised. Given the linearity of the Demand curves, we will obtain a quadratic profit function. Since profit function is strictly concave and quasiconcave, we can ensure a unique maximum, i.e. a unique optimal quantity for the firm to produce. The above explaination is for a 'standard' game with Linear Demand function, say, $$max(a-bq,0)$$, with cost function, say, $$C_i=c_iq_i$$ for, say, a duopoly, $$q=q_1+q_2$$

What I need to understand is that if we have non-linear demand function, such that we cannot ensure a unique optimal quantity for profit maximisation (which would be the Best Response function), how would the Nash Equilibria or Equilibrium be obtained? That said, is it possible for a Cournot game, or for that matter, any game to have more than one Best Response?

I'm still a bit rusty on the concepts, please bear with me. Is what I'm trying to understand, right conceptually? What am I missing?

• Having more than one best response is not unusual at all. That is why we usually talk about a best-response correspondence (instead of a function). See, e.g., en.wikipedia.org/wiki/Best_response – Bayesian Apr 23 '19 at 9:16

In general a best response function returns a set of best responses. This can be seen in much simpler games than Cournot.

To give a degenerate example, if a player is always indifferent between their strategies, their best response function will always return the set of all their strategies.

When you have best response functions that give sets with more than one element, Nash equilibrium happens when each player's play is contained in the other player's best response set.

• I think it's more appropriate to use best response correspondence if it returns multiple best responses to a particular strategy (profile), and reserve best response function to refer to single-valued best responses. – Herr K. Apr 22 '19 at 16:53

"we cannot ensure a unique optimal quantity for profit maximisation"
Yes, it is possible you can have two( or more) points as the maximum, then both of them will be the best responses.

Best Response Function, (BRF) is the function matching the opponent's decision to my best response. BRF doesn't have to be in a neat form. It can be expressed as a table or a piecewise function.

Nash Equilibrium is the point where both players best responses meet. It should be optimal against optimal. If not, it means someone can be better off by changing his/her strategy.