Terminal Games with 3 Players and 3 Terminals
Our “Typological Sort” method applies the following two steps:
1.
If a player can reach his optimal terminal directly from any node then merge that node
into the terminal.
2.
If any node can reach only terminals then merge it into its most preferred terminal.
Lemma 1: For any single application of move 1 or move 2 on any Node X on a two player game, if the
original graph contained a Nash Equilibrium, there is some play on the resulting graph that results in a
Nash Equilibrium. This play is also a Nash Equilibrium on the original graph (with the edge from X to its
preferred terminal chosen).
Proof: We address two cases. In Case 1, X is a member of the Equilibrium path. Since X prefers the edge
specified in typological sort over any other edge, the equilibrium path must result in X’s preferred
terminal, otherwise X would switch (meaning the play didn’t have Nash Equilibrium). Hence, changing X’s
edge to that chosen by the typological sort would maintain the same terminal and therefore have no
effect on the Nash Equilibrium.
In Case 2, X is not a member of the Equilibrium path. Assume changing X to point to a terminal would
break the Nash Equilibrium. This would mean that there exists some node Y on the path that would
redirect towards X resulting in X’s new terminal. Y cannot be controlled by the same player as X,
otherwise that player could singlehandedly redirect to the new terminal and the old path couldn’t have
been in Nash Equilibrium. Hence both players prefer the new Terminal to the old terminal, and there
exists a Nash Equilibrium to Y, so changing X and Y to point to the terminal also results in a Nash
Equilibrium.
Problem: Tries to say too much, switches between original and new equilibrium. Only the new
equilibrium matters.
New Lemma 1: For any typologically sorted Terminal Game, the Nash Equilibrium of this game is also a
Nash Equilibrium on the original game (with the edges pointing towards the Terminals of the typological
sort).
Proof: Assume player 1 could redirect a node X in the typologically sorted section of the graph to
produce a better outcome. Then that edge provides a better outcome then the original edge,
contradicting the rules of typological sort which only choose the best possible outcome. Since all the
nodes in the terminal sections point to the same terminals in both graphs, there is Nash Equilibrium in
the non-terminal sections of one graph if and only if the same equilibrium exists in the other.
Method 0: We set the nodes controlled by player 3 as follows: If any node controlled by player 3 can
reach a terminal or a node with a temporary value, we give it the temporary value corresponding to its
preferred choice of those terminals/values. Once no more changes can be made we set the temporary
values of player 3’s nodes. For all nodes that result in no values, we point the node towards any path to
its preferred terminal (or a cycle if no path to a terminal exists).
Problem: Method 0, combined with Typological sort, may induce a cycle (See illustration).
New Method 0: Begin with player 3's most preferred terminal. Fix all 3 nodes that can reach that
terminal. Then move to the next preferred terminal, setting nodes in the same manner. Finally set all
nodes that can reach the third terminal.
This must be combined with typological sort, for a new method 1.
Problem: We may induce cycles or set 3 to a non preferred node once sorting is complete (see
illustration).
Newer Method 0: Begin with preferred terminal and alternate the two sorts until no more changes.
Switch to terminal 2. If alternating setting nodes to 2 results in a a changed node pointing to one, revert
all other changes and return to step one. Since the terminal area is monotonically growing, we will
eventually reach a point where no more changes can be made.
Problem: It may not be monotonically growing see expanded illustration. In fact, we can still induce a
cycle.
Lemma 2: Setting player 3’s edges in the prescribed manner will always allow a path to a terminal, unless
the graph contained no path to a terminal.
Proof: Since every node is set to point towards a terminal (wherever a path to terminal exists), the
resultant graph will include a path from the origin to terminals unless there is no such path.
Method 1: We alternate Typological Sorts for players 1 and 2 with Method 0, until both methods result
in no change. We then find the Nash Equilibrium for the two remaining players on the resultant graph.
Outside of the path from the origin to a terminal, we set all nodes that point towards non-preferred
terminals to point elsewhere.
Lemma 3: The graph resulting from Method 1’s application to any terminal game results in a Nash
Equilibrium when player 3’s choices are held constant.
Proof: As shown in Boros [2002], any terminal game with two players is tight and therefore includes a
Nash Equilibrium. Since player 3’s nodes are set we can treat any edge to a player 3 node as an edge to
the following node resulting in a normal, 2 player terminal game.
Lemma 4: Assume for any 2 player, 3 terminal game we have a Nash Equilibrium to a terminal node A.
Then by switching all nodes that point to terminals B and C to point elsewhere, we maintain the
equilibrium.
Proof: Assume not. There exists some route from a node X on the path to a node Y which, if switched
away from terminal B would allow player 1 to reach preferred terminal C (1, B and C W.L.O.G.). If player 1
controls Y, 1 controls a route all the way to preferred terminal C, so the current play is not in Nash
Equilibrium →←. If 2 controls Y, obvious 1 prefers A and C to B (otherwise he would link X to Y) hence the
last vertex in 1’s alternative route (which he must control, recall that one can single-handedly cause this
path with X’s switch) should have been merged with A via the topological sort →←. Hence, no such route
exists.
Theorem 1: Method 1 of play results in a Nash Equilibrium for all three players on the graph.
Proof: For players 1 and 2 the game is already in Nash Equilibrium (as shown in Lemma 3 and 4
{following the conclusions in Boros [2002] } ), meaning given the current play, neither player 1 nor 2 has
incentive to switch. Since no nodes now points to a different terminal than during the sorting phase of
method 1, and no player 3 node with direct access to a terminal has access to any preferred terminal,
player 3 cannot redirect the path to any preferred terminal. Therefore the graph is in Nash Equilibrium.
Citations
Boros E., Gurvich V., On Nash-solvability in pure stationary strategies of finite games with perfect
information which may have cycles. Section 3.1 Theorem 5; 3.2 Proposition 3.