[MUSIC] Now we are ready to mathematically formalize the multistage game. In order to do that, we need to divide the set of vertexes of the graph into the subset x_i, from 1 to n, and set x_(n+1). x_i is the set of personal positions of the player i. So these are the set of vertexes of the graph where player i makes a move. Set x_(n+1) is the set of final or terminal positions. On this set of vertexes, we will define the payoffs of the game. For our particular game model, set of players consists of two players, player one and player two. The set of personal positions of player one includes vertexes which are marked by the circles. The set of personal positions of player two includes vertexes that are marked by the squares. So as you can see on the graph, on the first stage, player one makes a move. On the second stage, if player one chooses the alternative one, then on the second stage in position x(2.1), player two makes a move, etc. The set of final or terminal positions are the positions on the edges of the graph, which are not the set of personal positions of the first player and which are not the set of personal positions of the second player. Next thing we need to do in order to define the game is to define the payoff of player i. So the payoff of player i is a real-valued function H_i(x) defined on the set of final positions x_(n+1). Next thing we need to do is to define a strategy of player i. Strategy of player i is a single valid mapping u_i which for each vertex from the set of personal positions of player i assigns the next vertex on the graph. For our particular game model payoffs for each player are defined on the set of final positions which, for example, if we are in the position x(1.3) and player 1 chooses the alternative one, then the payoffs are five and one. So the first player receives a payoff equal to five and the second player receives a payoff equal to one. So how the game evolves. On the first step player one in our particular game model makes a move. He has three alternatives. For example, he choose the alternative one. Then at the next position, x(2.1), player two makes a move. For example, he also chooses alternative one. Then at the position x(1.3), again, player one makes a move. And for example he choses the alternative one then the payoff for the first player is equal to five, the payoff for the second player is equal to one and then the game terminates. Multistage games with perfect information can be transformed to the games in normal form that we have already studied. How can we do that? We need to define the strategy profile. The strategy profile is the vector of strategies that the players use. Next thing you'll need to do is we need to define the payoff function of player i K_i which is a function of strategy profile. And we will define it as a payoff of player i in the final position which corresponds to the strategy profile that was chosen by the players. So at the beginning of the game, players choose strategies. So they choose the alternative for any vertex of the graph. This is very important. According to this strategy profile, the unique path is realized. For the unique path, there is a unique position from a set of terminal positions. And for the unique terminal positions we have the unique payoffs of the players. Now we can say that the multistage game with perfect information can be transformed to the game in normal form. Let's construct the game in normal form for the example that we have already considered. In here, the problem is that the first player would have 864 strategies, and the second player would have 576 strategies. And if we would want to define it as a bimatrix game, then we would not be able to analyze it in our own line of course. Now we also need to define the subgame of the game Gamma starting at the position z. As the initial game defined on the graph, starting in the positions z. So the only difference is that the game is defined on the subset of the set of the vertexes of the initial graph. On the slide, you can see the sub game, which starts at the position x(2.4), and the corresponding graph for the game is highlighted. On this slide you can see a list of references where you can get more information about the multistage games with perfect information and corresponding examples.