# Game theory applied to invoicing

There is a traditional practise in precuement of almost any kind of paying an invoice at l ast thirty days after it has been received.

Is there any application of game theory here that can justify this practise as the most efficient Nash equilibrium for the repeated game where either two players play pay/don’t pay and 30 days/or immediately.

Or maybe two players playing 30 days/immediately and work with in future / or not?

## 1 Answer

Consider one-shot 2-player game as given in the table, where not paying results in punishment of $-\infty$. Thus, not paying will never be played. Also if player pays at the end of the game (vs the beginning of the game) she gets some benefits, such investing the cash and receiving interests for the 30 days period at monthly interest rate $r \in (0,1)$. Conversely, if a player gets paid at the beginning of the period she can invest that money, but she cant do so if she gets paid at the end of the period. Finally, suppose $a>b>0$.

\begin{array}{|l||*{2}{c|}}\hline {} &{Pay\ Now}&{Pay\ Later}\\ \hline\hline Pay\ Now &\big[-a + b;(1+r)(-b+a)\big] &\big[-a + b;- (1-r)b + (1+r)a\big] \\ \hline Pay\ Later &\big[-(1-r) a + (1+r)b; -b + a\big] & \big[-(1-r) a + b,-(1-r) b + a\big] \\ \hline \end{array}

The unique Nash equilibrium of the game is (Pay Later, Pay Later). Regarding efficiency, the Nash equilibrium outcome is weakly the most efficient outcome (only paying now by both players results in an inefficient outcome) and is equal to $r(a+b)$.

Note: The source of inefficiency of paying now by both players is a consequence of an implicit assumption that total resources of players are greater than $a$.