Modified parcheesi game A "modified Parcheesi" game starts with the following position:

First $x$ flips a fair coin. If heads he can move two spaces or pass. If tails he can move one space or pass. If he occupies the other player's position, then the other player returns to start. Each player tries to reach the Home square, in which case he wins. If he "passes" the Home square he still wins. When one counts board positions and distinguishes among them by who moves there are $6$ positions. If there are $2$ consecutive passes the game is a draw. I am curious if there is anything in the game theory literature which refers to this question/gives an analysis of the game? Some variant of this game is talked about in Binmore's game theory textbook Playing for Real, but the analysis is a bit simplistic, nor are any references to any sources given...
 A: Suppose winning the game gives you a payoff of $1$, and losing gives you a payoff of $0$. There are $3$ board positions as below, and call them $A$, $B$, and $C$ respectively, when it is $x$'s turn to play, and call them $D$, $E$, and $F$ when it is $y$'s turn to play.
             
                     
         
Let $a$ be the value of $A$ for player $x$, $b$ be the value of $B$ for player $x$, and so on. Since the game is zero sum, we have $d = 1 - a$, $e = 1 - c$, and $f = 1 - b$. With some abuse of notation, simulating some moves yields following figures.
                                 
We see immediately from the figure in the middle that $b = 1$, and thus $f = 0$. The figure in the right tells us that the following equation must hold: $$c = {1\over2} \cdot 1 +{1\over2}\max\{f, e\} = {1\over2} + {1\over2}(1 - c),$$and thus $c = 2/3$ and $e = 1/3$. Let us see what happens in the figure in the left:$$a = {1\over2} \cdot 1 + {1\over2}\max\{d, e\} = {1\over2} + {1\over2}\max\left\{1 - a, {1\over3}\right\},$$and thus $a = {2\over3}$ and $d = {1\over3}$. To wrap up, $x$ moves whenever he gets heads, and moves when he gets tails, except in the beginning of the game, where he is indifferent between moving one step and passing. Similar result applies to $y$.
If one was interested in a variant of this game where passing over Home takes you back to start, one just builds on the analysis we have above, except now having heads does not necessarily mean a win.
