Applied mathematics
Equilibrium in the problem of choosing the meeting time for $N$ persons
V. V. Mazalovab,
V. V. Yashinb a St Petersburg State University, 7–9, Universitetskaya nab., St Petersburg, 199034, Russian Federation
b Institute of Applied Mathematical Research, Karelian Research Centre of Russian Academy of Sciences, 11, ul. Pushkinskaya, Petrozavodsk, 185910, Russian Federation
Abstract:
A game-theoretic model of competitive decision on a meet time is considered. There are
$n$ players who are negotiating the meeting time. The objective is to find a meet time that satisfies all participants. The players' utilities are represented by linear unimodal functions
$u_i(x), x\in[0, 1], i=1,2,...,n$. The maximum values of the utility functions are located at the points
$i/(n-1), ~i=0,...,n-1$. Players take turns $1 \rightarrow 2\rightarrow 3 \rightarrow \ldots\rightarrow (n-1) \rightarrow n\rightarrow 1\rightarrow\dots ~. $ Players can indefinitely insist on a profitable solution for themselves. To prevent this from happening, a discounting factor
$\delta<1$ is introduced to limit the duration of negotiations. We will assume that after each negotiation session, the utility functions of all players will decrease proportionally to
$\delta$. Thus, if the players have not come to a decision before time
$t$, then at time
$t$ their utilities are represented by the functions
$\delta^{t-1}u_i(x), ~i = 1, 2,..., n.$ We will look for a solution in the class of stationary strategies, when it is assumed that the decisions of the players will not change during the negotiation time, i. e. the player
$i$ will make the same offer at step
$i$ and at subsequent steps
$n+i, 2n+i, \ldots$ . This will allow us to limit ourselves to considering the chain of sentences $1 \rightarrow 2 \rightarrow 3 \rightarrow \ldots \rightarrow(n-1) \rightarrow n\rightarrow 1.$ We will use the method of backward induction. To do this, assume that player
$n$ is looking for his best responce, knowing player
$1$'s proposal, then player
$(n-1)$ is looking for his best responce, knowing player
$n$'s solution, etc. In the end, we find the best responce of the player
$1$, and it should coincide with his offer at the beginning of the procedure. Thus, the reasoning in the method of backward induction has the form $1 \leftarrow 2\leftarrow 3\leftarrow \ldots\leftarrow(n-1)\leftarrow n\leftarrow 1.$ The subgame perfect equilibrium in the class of stationary strategies is found in analytical form. It is shown that when
$\delta$ changes from
$1$ to
$0$, the optimal offer of player
$1$ changes from
$\frac{1}{2}$ to
$1$. That is, when the value of
$\delta$ is close to
$1$, the players have a lot of time to negotiate, so the offer of player
$1$ should be fair to everyone. If the discounting factor is close to
$0$, the utilities of the players decreases rapidly and they must quickly make a decision that is beneficial to player
$1$.
Keywords:
optimal timing, linear utility functions, sequential bargaining, Rubinstein bargaining model, subgame perfect equilibrium, stationary strategies, backward induction.
UDC:
519.8
MSC: 91B26,
91A55 Received: August 8, 2022Accepted:
September 1, 2022
DOI:
10.21638/11701/spbu10.2022.405