Equilibrium in the problem of choosing the meeting time for $N$ persons

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 $1$.