Monday, 22 April 2013

the extensive form of a game illustrated reference watson, joel 2004.

The Extensive Form


The model which is used to illustrate the extensive form game is based on a story adapted from a game-theory text, which itself borrows it from a real-life case.(Watson, 2004) However this story suits our purpose which is to illustrate theoretical concepts of game theory relevant to agricultural markets.

The story is about animated films on bugs. The setting is the fall of 1998, when two films about bugs were among the many animated films to release in that period: Disney Studio’s A Bug’s Life and Dreamwork SKG's Antz. What was followed with great interest was the rivalry that brought the twin films into head-to-head competition.

This rivalry developed in the following way: Rumor has it that Disney executives pondered the idea of an animated bug movie in the late 1980s, at the time at which Jeffrey Katzenberg's was in charge of the company's studios. However, A Bug's Life was not conceived until after Katzenberg left Disney in a huff over not given a promotion. Katzenberg resigned in August 1994, shortly thereafter a Bug’s life was pitched by Pixar Animation to Disney, the proposal was accepted by Disney Boss Michael Eisner  and the film went into production.

About this time Steven Spielberg and David Geffen joined Katzenberg to form Dreamwork SKG, a new study with great expectations. Then in 1995 SKG joined computer animation firm PDI to produce Antz. It is possible that only after the two studios decided to produce these films did they learn about each other's choices. It was then that a series of calculated strategies and actions took place.

First of all, Disney chose to release A Bug’s Life in the 1998, Thanksgiving season, when SKG's Prince of Egypt was originally scheduled to open in theaters. In response, SKG decided to delay the release of Prince of Egypt until the Christmas season and rushed completion of Antz so that it could open before A Bug's Life and claim the title of "first animated bug movie."

This story is interesting because of the larger than life characters, the legal issues (did Katzenberg steal the bug idea from Disney?), and the complicated business strategy. In addition, there were the bad feelings generated. Katzenberg sued Disney for not paying bonuses due to him. Eisner had to admit in court, much to his embarrassment,  that he may have said of Katzenberg, “I hate that little midget” . Rumour was that Pfizer CEO Steve Jobs (also of Apple fame) believed Katzenberg stole the bug movie concept.

Now we stylize the story of the bug films into a mathematical model.

In the rather dull case in which Katzenberg makes a decision to stay at Disney (the lower branch from node a), assume that the game is over. On the other hand, if Katzenberg decides to leave, then other decisions need to be made. First, Eisner must decide whether to produce A Bug's Life. Figure 1.2 shows how the tree is expanded to include Eisner's choice. Note that, because Eisner has to make ns decision only if Katzenberg has left Disney, Eisner's move occurs at node b. Eisner’s two options — produce A Bug's Life or not — are represented by the two branches leading from node b to two other nodes, c and d.

After Eisner decides whether to produce A Bug's Life, Katzenberg must choose whether to produce Antz. Katzenberg's decision takes place at either node c or node d, depending on whether Eisner selected produce or not, as depicted in Figure 1.2. Note that there are two branches from node c and two from node d. Observe that Katzenberg's initial is placed next to c and d, because he is on the move at these nodes.
Figure 1.3

At this point, we need to introduce a critical matter: information. Specifically, does the tree specified so far properly capture the information that players have when they make decisions? With the extensive form, we can show the players' information by describing whether they know where they are in the tree as the game progresses. For example, when Katzenberg decides whether to stay or leave, he knows that he is making the first move in the game. In other words, at node a Katzenberg knows that he is at node. a. Also, because Eisner observes whether Katzenberg stays or leaves, when Eisner has to decide whether to produce A Bug's Life he knows that he is at node b.
However, as the story indicates, each player has to select between producing or not producing without knowing whether the other player has decided to produce. In particular, Katzenberg must choose whether to produce Antz before learning whether Eisner is producing A Bug's Life. The players get to know about each other's choices only after both are made.

Referring again to the tree in Figure 1.2, we represent Katzenberg's lack of information by speci­fying that, during the game, he cannot distinguish between nodes c and d. In other words, when Katzenberg is on the move at either c or d, he knows that he is at one of these nodes but he does not know which one' Figure 2'3 captures this lack of information with a dashed line connecting nodes c and d. Because these two nodes are connected with the dashed line, we need to label only one of them with Katzenberg's initial.

Assume that, if either or both players chose not to produce his film pro­posal, then the game ends. If both players opted to produce, then one more decision has to be made by Katzenberg: whether to release Antz early (so it beats A Bug’s Life to the theaters). Adding this decision to the tree yields Fig­ure 1.4, Katzenberg makes this choice at node e, after learning that Eisner decided to Produce A Bug's Life.
Figure 1.4
Figure 1.4 describes all of the players' actions and information in the game. Nodes a, b, c, d, and e are called decision nodes, because players make decisions at these places in the game. The other nodes (f, g, h, I, m, and n) are called terminal nodes; they represent outcomes of the game—places where the game ends. Each terminal node also corresponds to a unique path through tree, which is a way of getting from the initial node through the tree by following branches in the direction of the arrows. In an extensive form, there is a one-to-one relation between paths and terminal nodes.

It is common to use the term information set to specify the players' information at decision nodes in the game. An information set describes which decision nodes are connected to each other by dashed lines (meaning that a player cannot distinguish between them). Every decision node is contained in an information set; some information sets consist of only one node. For example, the information set for node a comprises just this node — Katzenberg can distinguish this node from his other decision nodes. Nodes b and e also are their own separate information sets. Nodes c and d, however, are in the same information set.
The information sets in a game precisely describe the different decisions that players have to make. For example, Katzenberg has three decisions to take in the game at hand: one at the information set given by node a, another at the information set containing nodes c and d, and a third at the information Set shown by node e. Eisner has one decision to make—at node b. Remember that only one decision is made at each information set. For example, because nodes c and d are in the same information set, Katzenberg makes the same choice at c as he does at d (produce or not). We always assume that all nodes in an information set are decision nodes for the same player.
You should check that the tree in Figure1.4 delineates game elements I through 4 noted 1)  A list of players
2)  A complete description of what the players can do( their possible actions.)
3) A description of what the players know when they act
4) A specification of how the players actions lead to outcomes.
We have just one more element to address: the players' preferences over the outcomes. To understand preferences, we must ask questions such as, Would Katzenberg prefer that the game end at terminal node f  rather than at terminal node l ? To answer such questions, we have to know what the players care about. We can then rank the terminal nodes in order of preference for each player. For example, Eisner may have the ranking h, g, f, n, l, m; in words, his favourite outcome is h, followed by g, and so on. It is usually most convenient to represent the players' preference rankings with numbers, which are called payoffs or utilities. Larger payoff numbers signify more preferred outcomes.
For many economic games, it is reasonable to suppose that the players care about their monetary rewards (profit). Katzenberg and Eisner probably want to maximize their individual monetary gains. Therefore, let us define the payoff numbers as the profits that each obtains in the various outcomes. The profits in this example are purely fictional – who knows what a film’s true profit is? For example, in the event that Katzenberg stays at Disney, suppose he gets $35 million and Eisner gets $100 million. Then we can say that node n yields a payoff vector of (35,100). This payoff vector, as well as payoffs for the other terminal nodes, is recorded in Figure 1.5. Note that, relative to Figure 1.4, the terminal nodes are replaced by the payoff vectors. In addition, we use the convention that one player’s payoff is always listed first. Because Katzenberg is the first player to move in this game, his payoff is listed first by us.
Figure 1.5 depicts the full extensive form of the Katzenberg-Eisner game; it represents all of the strategic elements. A more compact representation (with all actions abbreviated to single letters) appears in Figure1.6. This compact representation is usually the style in which I will present extensive forms to you. Observe that one of Katzenberg’s action choices s labelled N’ to differentiate it from the other “Not” action earlier in the tree. Then given the name of a player and the name of an action, you will be able to figure out to where in the tree is being referred. It will be important that you differentiate actions in this way whenever you draw an extensive form game. Incidentally, you should always maintain conformity in labeling branches from nodes in the same information set. For example, Katzenberg’s choices regarding whether to produce Antz are represented by actions P and N in Figure 1.5.
Figure 1.5.
The labels P and N are each used for two branches – from nodes c and d, which are in the same information set (refer to Figure1.5). When Katzenberg finds himself at either node c or at node d in the game, he knows only that he is at one of these two nodes and has to choose between P and N. In other words, this information set defines a single place where Katzenberg needs to make a decision.

In the end, Antz led to a bit more than $90 million in revenue for Dreamwork SKG, whereas A Bug's Life produced more than $160 million. These films probably each cost about $50 million to produce and market, meaning that Katzenberg and Eisner faired pretty well. But did they make the "right" decisions? It may not be possible to answer this question fully by using the same tree just developed but analysis of our extensive form will some insights that are useful for instructing Katzenberg, Eisner, and other game players in how best to play. In addition, designing and analyzing game models will help understand a wide range of strategic issues. Let us not ignore the possibility that you may be advising the likes of Katzenberg and Eisner someday.



No comments:

Post a Comment