I consider the following game : three players have a pie to share. Player 1 proposes a division and then simultaneously player 2 and 3 respond either yes or no.
Each player prefer to have more than less share and the payoff of each player is the corresponding share that is adopted.
This game can be naturally represented by a tree (extensive form game).
I would like to find the subgame perfect equilibrium using the usual backward induction.
The tree is as follows : player 1 proposes a triplet $(x,y,z)\in[0,1]^3$ (res presenting the share of each player starting from player 1 to player 3), then player $2$ responds yes or no and then player $3$ responds yes or no.
For player $3$ to accept we need to have $z>0$ and for player $2$ to accept we need to have $y>0$.
Then, for player $1$ the choice is more difficult I find. Propose $(x,y,z)$ with each component strictly positive seems to be the « natural choice » however how to define precisely the quantity ?
The fact that each player prefer to have more share than less (especially the first player, which chooses the quantity) make me doubt about finding a subgame perfect equilibrium of the form $\{ (x,y,z),\text{yes},\text{yes}\}$ with $x,y$ and $z$ strictly positive.
I would like to have your help on this question please.
Thank you a lot
Some precision : For the proposition of share of the player 1 to be adopted, we need to have the agreement of player 2 and 3, otherwise the share is not adopted and everybody get $0$.