Reaction function

In game theory, a reaction function is a mathematical function that indicates which strategy a player should optimally choose, given the (observed or expected) strategy of the other player (s). In this way, for example, reaction functions model the reactive behavior of oligopolistic providers ( oligopoly ) in price theory .

Reaction functions as best response functions (best-response function or best-reply function) or response functions referred to (see The best answer ).

Although the term reaction function is common, it is somewhat misleading. A reaction in the true sense of the word is not even possible for the company, because the choice of the counterparty - regardless of whether it relates to its offer quantity or price - is unknown when making its own decision.

example

Reaction functions in the example

Consider the following simple game: In a small ancient town there are exactly two fishermen, each with a boat; There is no way of storing caught fish, which means that every fish caught is either sold in the morning or has to be destroyed. In addition, there is always enough fish in the sea. The price for every fish sold is formed every morning in the market, and the fishermen are aware that the unit price is lower, the more is offered in total on the market. In fact, over time, the fishermen come to the conclusion that the price p depends on the total quantity q _A + q _B offered as follows : p = 40 - (q _A + q _B ). The two fishermen are differently efficient in their work: Fisherman A needs three times as long to catch a fish as fisherman B and is therefore three times as high as B (6 instead of 2 monetary units).

The following functions represent the profit of A and B depending on the amount offered:

${\displaystyle \pi _{A}\left(q_{A}\right)=(p-6)\cdot q_{A}=\left[\left(40-q_{A}-q_{B}\right)-6\right]\cdot q_{A}=34q_{A}-q_{A}^{2}-q_{A}q_{B}}$

${\displaystyle \pi _{B}\left(q_{B}\right)=(p-2)\cdot q_{B}=\left[\left(40-q_{A}-q_{B}\right)-2\right]\cdot q_{B}=38q_{B}-q_{B}^{2}-q_{A}q_{B}}$

Note that the supply of the other is always assumed to be exogenously given here; the two fishermen can only decide on their own amount. They also choose this every morning without knowing the amount chosen by their competitor (simultaneous selection). Each fisherman now maximizes his profit, given the amount offered by the other fisherman (this consideration is initially not particularly meaningful due to the ignorance of this amount). However, it leads to the following first-order conditions:

For a: ${\displaystyle \pi _{A}'(q_{A})=0\Rightarrow 34-2q_{A}-q_{B}=0\Rightarrow q_{A}^{*}=17-0.5q_{B}}$

For B: ${\displaystyle \pi _{B}'(q_{B})=0\Rightarrow 38-2q_{B}-q_{A}=0\Rightarrow q_{B}^{*}=19-0.5q_{A}}$

The basic idea of ​​a reaction function is to write this optimality condition as a function of the quantity chosen by the competitor. There are then

${\displaystyle BR_{A}(q_{B})=17-0.5q_{B}}$

${\displaystyle BR_{B}(q_{A})=19-0.5q_{A}}$

the respective reaction functions of A and B. They indicate which set a profit-maximizing player should choose, given the selected set of the other player. Analogous to the above remark, one could assume that this formulation is again not meaningful, because the quantity chosen by the competitor is not known at all; the other's supply volume can also be interpreted, for example, as an assumption about the actual supply volume.

In this reading, the reaction functions are particularly suitable, for example, for visualizing the finding of the Nash equilibrium in this game. In Nash equilibrium, both players play the best mutually best responses. However, this means nothing other than that every combination of quantities (q _A , q _B ) which is equilibrium in the Nash sense must lie on the reaction function of A and B - graphically, this means that the equilibrium is an intersection of the both reaction functions must act. In the above example, the (only) Nash equilibrium could therefore easily be determined by inserting one equation into the other (here q _A = 10 and q _B = 14).

Best-answer correspondence

One speaks instead of a reaction or best answer function from a best answer correspondence (best-reply correspondence), if a strategy profile of the other players exists on the number of strategies (or actions here) a best answer is . For example, in a simultaneous game with the payoff structure, reads

Player 1 / Player 2	a	b
A.	-4; 3	2; 5
B.	-4; -4	-8th; 2

the best-answer correspondence of player 1 (depending on the action of his opponent) ${\displaystyle s_{2}}$

${\displaystyle BR_{1}(s_{2})={\begin{cases}\{A,B\}&\mathrm {falls} \;s_{2}=a\\\{A\}&\mathrm {falls} \;s_{2}=b\end{cases}}}$.

Individual evidence

1. ↑ Response function - definition in the Gabler Wirtschaftslexikon.
2. ↑ Gernot Sieg: Game Theory . Oldenbourg Wissenschaftsverlag; Edition: 3 (October 6, 2010). ISBN 978-3486596571 . Page 14.
3. ↑ Harald Wiese: Decision and game theory . Springer Berlin Heidelberg; Edition: 2002 (January 1, 2002), ISBN 978-3540427476 , page 117.
4. ↑ Ulrich Blum: Applied Institutional Economics . Gabler Verlag; Edition: 2005 (January 1, 2005). ISBN 978-3409142731 . P. 62.
5. ↑ The example follows (slightly modified) Avinash Dixit and Susan Skeath: Games of Strategy. 2nd ed. WW Norton, New York 2004, ISBN 0-393-92499-8 , pp. 147 f.