Sequential game tree

The entries of the cost matrix, 9. Every nonleaf vertex is called a decision vertex : One player must select an action. There are two possible interpretations of the game depicted in Figure This does not correspond to the zero-sum game formulation introduced in Section 9. In this case, it is not equivalent to the game in Example 9.

This is equivalent to assuming that both and make their decisions at the same time, which is consistent with Formulation 9. The tree could have alternatively been represented with acting first. Now imagine that and play a sequence of games. A sequential version of the zero-sum game from Section 9. This will model the following sequential game : Formulation A stage as considered previously is now stretched into two substages , in which each player acts individually.

It is usually assumed that always starts, followed by , then again, and so on. The end positions are those vertices with now outgoing arcsat them the payoffs for each player have to be noted.

The whole directed graph with all labels is called the Game Digraph or Extensive Form of the sequential game. But note that we need to extend the definition further later , when we take randomness and non-perfect information into account. The Extensive Forms in the previous two examples are so-called Game Trees.

Let's avoid the formal definition and just say that a Game Tree looks like a tree, rotated by 90 or degrees. Trees arise if one considers a position to be the whole sequence of previous decisions made by all the players who have moved so far. But in the previous example, it is also obvious that there is some redundancy. Why do we have two "3w" positions? Granted, both have a different history, one resulting from position "5b" with Black taking two stones, and the other from position "4b" with Black taking one stone.

But that is not relevant for the future, as can be seen in the Game Tree by the fact that both subtrees hanging at the corresponding vertices are identical. So if White decides how to play in these two positions, he or she should come to the same conclusion.

If we identify such corresponding positions in the Game Tree of the previous Nim 6 example, we get the following Extensive Form:. It should have become clear from the previous discussion that games may have different descriptions by Extensive Forms. We will see more examples later. Note also that in the literature mostly game trees are used to describe extensive forms. Our approach of game digraphs has the advantage of reducing the number of positions.

Next we will discuss a simple and very powerful method how to analyze sequential games. However, this method works only for so-called "finite games". A sequential game is finite if it has a game tree with finitely many vertices. A player moves by raising one or two fingers. A player loses when raising the same number of fingers than the other player in the previous move. Then the payoffs are -1 for the loser and 1 for the winner.

How do you play this simple zero-sum game? Who will win? Obviously nobody will lose, since losing can very easily be avoided. So if nobody loses, nobody wins.

The two players will continue playing forever. The Game Tree goes on and on, indicated by the dashed line, and is therefore infinite. Thus the game is not finite. We can also use a Game Digraph.

Then a non-end position is uniquely determined by who is about to move, White or Black, and how many fingers were last shown. So we have four of these positions, labeled as W1, W2, B1, and B2. Of course there is one more position, the start position with White to move, where no fingers have been shown yet, and two end positions, one where White wins and one where Black wins.

Why does the previous example have a finite game digraph although the game is not finite? The reason is that the game digraph is cyclic. That means that it is possible to follow arcs from some vertex x and reach the vertex x again after some time, like in W1, B2, W1 in the above example. Games with a cyclic game digraph have always an infinite game tree, but a sequential game is finite if it has some acyclic not cyclic finite game digraph.

Why does the previous example have a finite Game Digraph although the game is not finite? The reason is that the Game Digraph has cycles. It is possible to follow arcs from some vertex x and reach the vertex x again after some time, like in W1, B2, W1 in the above example. Thus a sequential game is finite if it has some acyclic finite Game Digraph. There are theoretical reasons why we exclude these infinite games, but of course there are more practical reasons: We need to have a guarantee that a game eventually ends.

Note that even chess would not be finite, were it not for the "move rule" that says that a chess play ends as draw if no pawn has been moved within 50 moves. What players may want from Game Theory is advice how to play. In a sequential game that means that for every position that is not an end position the player who has to move there would need advice about which of his or her possible options to choose.

Such a list of recommendations for all the player's positionseven for those positions for which we are rather certain that they will never occur in a play is called a pure strategy for that player. In this section we will present a procedure generating pure strategies for all players. Moreover, at each vertex some numbers will be attached by the procedure: the likely payoffs for each one of the players. The word "likely" has nothing to do with probabilities, which will only be discussed in the next chapter.

Its meaning here will be discussed later. Let us explain the method first at another example, this time a non-zero sum game with three players:. In the sequential variant considered, A has to vote first, then B, then C, and all votes are open. This is a variant of a game described in [Kaminski]. The Game Tree of this game is shown to the right. The behavior of legislator C is easiest to explain first, since at the time when C has to move, A and B have already moved.

So C doesn't need to anticipate their strategies. C may face four different situations, corresponding to the four vertices to the right of the tree.

In the first situation, when A and B have already voted for a raise, C can keep face by voting against the raise but still get the benefit of it. The same is true if A and B both rejected the idea of a raise and it is decided already. It is different if one voted for and one against a raise. Then C's vote counts. Since the money is more important than face for our fictitious legislators only! C would vote for a raise in that case. Now everybody can do this analysis, therefore if B has to decide, she can anticipate C's reactions.

If A voted for a raise, B knows that C will vote for a raise if she votes against, and conversely. So why not give C the burden of having to vote for the raise? Obviously B will vote against a raise in this situation. If on the other hand A voted for a raise, then B has to vote for a raise to keep things open. Since A knows all this too, A will vote against a raise, B will vote for it, and C will vote for it as well.

A has the best role in this game. Going first is sometimes useful, after all! This method of attaching a recommendation and "likely" payoffs for all players to all vertices of the Extensive Form, starting at the vertices "late" in the game and moving backwards is called "backward induction". We will explain it more formally below. Note that when a player accepts these decision found by backward induction, he or she must assume that the other players will also stick to these backward induction recommendations.

And why wouldn't they? Well, one reason could be that they are not able to analyze the game, since the game is maybe too complicated. Or a player can analyze the game, but doubts that all others can do so. In NIM 6 we can immediately assign "likely" payoffs to the vertices "1w" and "1b". In both cases, the player to move has just one option. In the "1w" case, White will move, take the remaining stone, and arrive at "0b", which is loss for Black, therefore a win for White.

Therefore White will win when facing "1w", and the expected payoffs are 1 for White and -1 for Black. In the same way, the likely payoffs at "1b" are -1 for White and 1 for Black. Having now values assigned to four vertices, we consider the vertices "2w" and "2b" whose successors all have values already. In the "2w" position, White can proceed to "0b" and win, and will do so, therefore the values there are 1 and -1 for White and Black.

Similar, at the position "2b" the likely payoffs are -1 and 1 for White and Black. Next we look at the positions "3w" and "3b", since again all their successors have been treated already. From "3w", White can move to positions "2b" or "1b" by taking one or two stones. But both these positions are unfavorable for White having both a likely payoff of -1 for White attached.

Therefore it doesn't matter what White chooses in this situation, and the likely payoffs at vertex "3w" are -1 and 1 for White and Black. The position "3b" is analyzed in the same way. Since from position "4w" positions "3b" and 2b" can be reached, but the first one is favorable for White and the second one is favorable for Black, White will choose the first option. This means that the likely payoffs of position "3b" are copied to position "4w".

In the same way, Black will move to position "3w" from position "4b", and the likely payoffs of position "3w" are copied to position "4w". We proceed in this way, and eventually arrive to assign likely payoffs even for the start vertex.

This means that at the very beginning, White expects to lose and Black expects to win. Procedure for Backward Induction : Your goal is to have likely payoff values for each player assigned at every vertex. As long as not all vertices have such values attached, do the following:.

There is still one problem in this procedure. What do we have if there are ties, if two or more successor vertices of a vertex have the same likely payoff for the player to move at that vertex.

This case will be discussed in Section 3. In , Ernst Zermelo showed that finite perfect-information sequential games can always be analyzed using backward induction. Actually he referred to chess, but this result can easily be generalized to the following:.

Theorem [E. Zermelo ]: Every finite perfect-information sequential game without random moves can be analyzed using backward induction. The likely payoff values of the start position are what the players should expect when playing the game provided all play rationally. This is one of the few proofs that will be covered. Students should understand the notion of a proof, and how the proof explains the construction.

Proof: Induction over the number of states. By construction, the outcome generated by backward induction obeys the following property, which of course has some conncetion to the Nash equilibrium definition:. Fact: If all players play their backward induction strategy, and if one single players deviates from his or her backward induction strategy, then this player will not get more as payoff.

Historical Remark 1: Ernst Zermelo was a German mathematician. He is nowadays mostly known for his work in the foundation of set theory, which itself has been invented by Georg Cantor about 30 years earlier. However, his paper on chess and the procedure that is nowadays called "backward induction" was his only contribution to Game Theory. In fact, Game Theory did not exist yet at that time. The idea underlying backward induction seems rather natural.

It is not known to me whether Zermelo was the first to express it. However, the idea has been reinvented several time. Although the backward induction solution seems to be the best the players can do in many situations, there are certain sequential games where players tend to play differently:.

A player to move has the choice either to pass, in which case both stacks grow slightly and the other player faces them, or to take the larger stack, in which case the other player gets the smaller one and the game ends. The game ends anyway after a fixed and known number of rounds, in this example after six rounds.

This game was introduced by Rosenthal [Rosenthal ] in the rounds version. It has a unique backward induction solution: Player 1 would take the money in the first round and end the game. On the other hand, players can both increase their payoff if they both wait much longer. The research being done by the University of Mississippi will not be limited to the medical marijuana industry however. The study is currently ongoing, so the possibilities for CBD products are endless.

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It can even be used for aromatherapy and skin care treatments as well. In sequential games, a series of decisions are made, the outcome of each of which affects successive possibilities.

Sequential games are represented through decision trees , with successive nodes at each decision point:. The game represented in this decision tree shows firm 1 choosing whether to compete in a monopolistic market or not.