Basics of game theory 1 (10/1/17 to 2/2/17) 5 What is a game? We wish to model real situations using games. So first we look at the issues that must be considered when the model is being constructed. We use the words agent and player interchangeably. a) The number of agents (or player groups) in a game. Each game must have at least two players. b) Information availability: Whether all players in the game have full and complete information when it is their turn to play. These are called games of perfect or complete information. For example, chess is such a game since both players know all the moves made by the other and the rules of the game (known to both players) constrain what moves can be made at each point in future. Games where players must move simultaneously (so or where some information is not available as in many card games are examples of games with imperfect information. This applies largely to sequential games where players move in turn. c) Moves or actions available to each player. This is normally specified in the form of a set of strategies for each player. Informally, a strategy is a rule or algorithm which tells a player what move to make at each point. d) The rules of the game. These are normally expected to be known to all players but in real life applications that are modelled as games one or more players may not be fully aware of the rules. e) Games can be one-shot or repetitive or sequential. A one-shot game is played only once. A repetitive game is one where a game is played repeatedly between the same players. A game may be finitely repeated or repeated infinitely. Player strategies/behaviour can vary depending on whether a game is one-shot or repetitive. A sequential game is one where players take turns to play and the game typically lasts over multiple rounds of play (e.g. chess). A sequential game can also be repetitive if it is played again and again. f) Presence of stochastic or random element. A player called `Nature' is invented when there is a stochastic or random element enters into the game. For example, dealing cards or throwing dice. In such cases a probability distribution is defined over all possible moves/states to specify the random move played by Nature. g) Nature of payoff. The goal of any player is to maximize his/her payoff or alternately win the game. player (or some times the group). For any game a payoff function for each player has to be defined. The domain of the payoff function is a set of strategy profiles (informally if the game has n players a strategy profile is the n-tuple th (s1; : : : ; sn) where si is a strategy from the strategy set of the i player) and the range is R. h) Cooperation: in some games, called cooperative games, one or more agents may cooperate with another agent or group. i) Static versus dynamic: there are two senses in which games can be dynamic. Games with incomplete information (e.g. when a player does not know the payoff or costs or other information that is private to other players) are dynamic since more information may become available as a game proceeds thereby changing available or chosen strategies. In the second sense evolutionary games are dynamic as strategies that have lower fitness gradually vanish from the population (see section 4 in lec. 1). 6 An example game Consider an example game described below: Two Langurs L(big) and l(small) are below a fruit tree that has 10 fruits on it. One or both of them must climb an shake the tree so that the fruit drop to the ground after which they can them. 3 Figure 1: L plays first. Figure 2: l plays first. Figure 3: L, l play simultaneously. Here is some more data on the game: {L takes 2 units of fruit energy to climb and shake the branches. {l (being light and small) takes 0 units of fruit energy for the same. { If L climbs then fruits are shared L-6, l-4. { If l climbs then fruits are shared L-9, l-1. { If both L and l climb then fruits are shared L-7, l-3. The goal of each langur is to get maximum net fruit energy units (1 fruit ≡ 1 fuit energy unit). Two moves are available to both Langurs. Either climb (c) or wait below (w). There are 3 possible scenarios {L moves first. {l moves first. { Both move simultaneously. Note that neither knows what move/strategy the other will play. We represent the game as a game tree. Figures 1, 2 are game trees when L l play first respectively and 3 is a tree when both play simultaneously. In figure 3 L moves first but it can also be drawn with l moving first - it is symmetric with respect to the two players. Figures 1 and 2 are perfect information games since both players know everything that the other player does. Figure 3 is an imperfect or incomplete information game since neither knows which strategy/move will be played by the other. So, the c and w nodes are connected with a dashed line to form what is called an information set. In figure 3 l does not know which of c or w L has played before making its own move since both play simultaneously. The nodes in an information set are indistinguishable for the player whose turn it is to play. A single node is also an information set - a set with just one element, namely the node itself. So, we can always talk in terms of information sets while talking about game trees. The game trees in figures 1 to 3 are finite game trees - that is they have finitely many nodes. We can also have infinite game trees where the tree has infinitely many nodes - for example a chess game where repetition of moves is allowed. A game tree may have infinite number of nodes but may simultaneously have finite depth. This can happen if one or more nodes have infinitely many children but the depth of a path from the root to any leaf node is finite. Such games are called finite horizon games. An example of such a game is where at a node a player's move corresponds to choosing any number from a real interval e.g. [a; b]; a < b; a; b 2 R. 4 P2 (column player l) P1 (row player L) [(c,c)(w,c)] [(c,c)(w,w)] [(c,w)(w,c)] [(c,w)(w,w)] c 5,3 5,3 4,4 4,4 w 9,1 0,0 9,1 0,0 Table 1: The normal form of the langur game. A game represented as a tree which shows the details of who moves when is called a game in extensive form. A game can also be represented as a table or matrix (n−dimensional where n is the number of players) the value of the ith dimension di is the number of strategies available to player i and the entries in the matrix give the payoffs for each player when it chooses that strategy. This is called the normal form of the game. For example the tree in figure 1 converts to the following normal form game. l's strategies say what l will play when L plays c and when it plays w. The game tree in figure 3 has the following normal form. Note that l has only two strategies since nodes in an information set are indistinguishable. P2 (column player l) P1 (row player L) c w c 5,3 4,4 w 9,1 0,0 6.1 Analysis of the game We assume both players are rational which means each player tries to maximize its payoff. Consider the game in figure 1 where L plays first. It reasons that playing w is best since the payoff is 9 if l climbs. But l must climb because then it gets at least 1 otherwise it gets 0. Note that L has just 2 strategies - w and c. However, l has 4 strategies. It can decide: i) play c irrespective of what L plays, ii) play w irrespective of what L plays, iii) play the same move that L plays, iv) play the opposite of what L plays. These 4 strategies are clear from the normal form of the game. To completely define a strategy for player i we have to give a rule/algorithm that gives the move for each information set when it is i0 s turn to play. For the game in figure 2 (l plays first) l can play w knowing that rationality demands L will play c to maximize its payoff resulting in a payoff of 4 for both. 6.1.1 Threats, promises Suppose for the game in figure 1 l threatens to play strategy iv) and makes L believe the threat then clearly L should play c because then its payoff is 4 which is better than 0 (if it plays w). This is usually called an incredible threat since the key question is how does l make L believe that it will play iv) since rationally w is the best strategy for L. Notice that if the threat does not work then both players lose. Similarly, l can make a promise that it will play c if L plays c.