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14.12 Game Theory Lecture Notes∗
Lectures 3-
Muhamet Yildiz†
In these lectures, we will formally define the games and solution concepts, and discuss the assumptions behind these solution concepts. In previous lectures we described a theory of decision-making under uncertainty. The second ingredient of the games is what each player knows. The knowledge is defined as an operator on the propositions satisfying the following properties:
- if I know X, X must be true;
- if I know X, I know that I know X;
- if I don’t know X, I know that I don’t know X;
- if I know something, I know all its logical implications.
We say that X is common knowledge if everyone knows X, and everyone knows that everyone knows X, and everyone knows that everyone knows that everyone knows X, ad infinitum.
1 Representations of games
The games can be represented in two forms: ∗These notes are somewhat incomplete – they do not include some of the topics covered in the class. I will add the notes for these topics soon. † Some parts of these notes are based on the notes by Professor Daron Acemoglu, who taught this course before.
- The normal (strategic) form,
- The extensive form.
1.1 Normal form
Definition 1 (Normal form) An n-player game is any list G = (S 1 ,... , Sn; u 1 ,... , un), where, for each i ∈ N = { 1 ,... , n}, Si is the set of all strategies that are available to player i, and ui : S 1 ×... × Sn → R is player i’s von Neumann-Morgenstern utility function.
Notice that a player’s utility depends not only on his own strategy but also on the strategies played by other players. Moreover, ui is a von Neumann-Morgenstern utility function so that player i tries to maximize the expected value of ui (where the expected values are computed with respect to his own beliefs). We will say that player i is rational iff he tries to maximize the expected value of ui (given his beliefs).^1 It is also assumed that it is common knowledge that the players are N = { 1 ,... , n}, that the set of strategies available to each player i is Si, and that each i tries to maximize expected value of ui given his beliefs. When there are only 2 players, we can represent the (normal form) game by a bimatrix (i.e., by two matrices):
1 \ 2 left right up 0,2 1, down 4,1 3, Here, Player 1 has strategies up and down, and 2 has the strategies left and right. In each box the first number is 1’s payoff and the second one is 2’s (e.g., u 1 (up,left) = 0, u 2 (up,left) = 2.)
1.2 Extensive form
The extensive form contains all the information about a game, by defining who moves when, what each player knows when he moves, what moves are available to him, and (^1) We have also made another very strong “rationality” assumption in defining knowledge, by assuming that, if I know something, then I know all its logical consequences.
A
B
C
D
is not a tree either since A and B are not connected to C and D.
Definition 3 (Extensive form) A Game consists of a set of players, a tree, an al- location of each node of the tree (except the end nodes) to a player, an informational partition, and payoffs for each player at each end node.
The set of players will include the agents taking part in the game. However, in many games there is room for chance, e.g. the throw of dice in backgammon or the card draws in poker. More broadly, we need to consider the “chance” whenever there is uncertainty about some relevant fact. To represent these possibilities we introduce a fictional player: Nature. There is no payoff for Nature at end nodes, and every time a node is allocated to Nature, a probability distribution over the branches that follow needs to be specified, e.g., Tail with probability of 1/2 and Head with probability of 1/2. An information set is a collection of points (nodes) {n 1 ,... , nk} such that
- the same player i is to move at each of these nodes;
- the same moves are available at each of these nodes.
Here the player i, who is to move at the information set, is asusumed to be unable to distinguish between the points in the information set, but able to distinguish between the points outside the information set from those in it. For instance, consider the game in Figure 1. Here, Player 2 knows that Player 1 has taken action T or B and not action X; but Player 2 cannot know for sure whether 1 has taken T or B. The same game is depicted in Figure 2 slightly differently. An information partition is an allocation of each node of the tree (except the starting and end-nodes) to an information set.
T B
x
L R L R
Figure 1:
1 x
T B
L R L R
Figure 2:
To sum up: at any node, we know: which player is to move, which moves are available to the player, and which information set contains the node, summarizing the player’s information at the node. Of course, if two nodes are in the same information set, the available moves in these nodes must be the same, for otherwise the player could distinguish the nodes by the available choices. Again, all these are assumed to be common knowledge. For instance, in the game in Figure 1, player 1 knows that, if player 1 takes X, player 2 will know this, but if he takes T or B, player 2 will not know which of these two actions has been taken. (She will know that either T or B will have been taken.)
Definition 4 A strategy of a player is a complete contingent-plan determining which
For instance in this game, player 2 knows whether player 1 chose Head or Tail. And player 1 knows that when he plays Head or Tail, Player 2 will know what player 1 has played. (Games in which all information sets are singletons are called games of perfect information.) In this game, the set of strategies for player 1 is {Head, Tail}. A strategy of player 2 determines what to do depending on what player 1 does. So, his strategies are:
HH = Head if 1 plays Head, and Head if 1 plays Tail; HT = Head if 1 plays Head, and Tail if 1 plays Tail; TH = Tail if 1 plays Head, and Head if 1 plays Tail; TT = Tail if 1 plays Head, and Tail if 1 plays Tail.
What are the payoffs generated by each strategy pair? If player 1 plays Head and 2 plays HH, then the outcome is [1 chooses Head and 2 chooses Head] and thus the payoffs are (-1,1). If player 1 plays Head and 2 plays HT, the outcome is the same, hence the payoffs are (-1,1). If 1 plays Tail and 2 plays HT, then the outcome is [1 chooses Tail and 2 chooses Tail] and thus the payoffs are once again (-1,1). However, if 1 plays Tail and 2 plays HH, then the outcome is [1 chooses Tail and 2 chooses Head] and thus the payoffs are (1,-1). One can compute the payoffs for the other strategy pairs similarly. Therefore, the normal or the strategic form game corresponding to this game is
HH HT TH TT Head -1,1 -1,1 1,-1 1,- Tail 1,-1 -1,1 1,-1 -1, Information sets are very important! To see this, consider the following game.
Game 2: Matching Pennies with Imperfect Information
1
2
Head
Tail
Head
Tail
Head
Tail
(-1, 1)
(1, -1)
(1, -1)
(-1, 1)
Games 1 and 2 appear very similar but in fact they correspond to two very different situations. In Game 2, when she moves, player 2 does not know whether 1 chose Head or Tail. This is a game of imperfect information (That is, some of the information sets contain more than one node.) The strategies for player 1 are again Head and Tail. This time player 2 has also only two strategies: Head and Tail (as he does not know what 1 has played). The normal form representation for this game will be:
1 \ 2 Head Tail Head -1,1 1,- Tail 1,-1 -1,
Game 3: A Game with Nature:
Nature
Head 1/
O
1
Left (5, 0)
Right (2, 2)
Tail 1/
O
2 Left
(3, 3)
Right (0, -5)
2.1 Dominant-strategy equilibrium
Let us use the notation s−i to mean the list of strategies sj played by all the players j other than i, i.e., s−i = (s 1 , ...si− 1 , si+1, ...sn).
Definition 7 A strategy s∗ i strictly dominates si if and only if
ui(s∗ i , s−i) > ui(si, s−i), ∀s−i ∈ S−i.
That is, no matter what the other players play, playing s∗ i is strictly better than playing si for player i. In that case, if i is rational, he would never play the strictly dominated strategy si.^2 A mixed strategy σi dominates a strategy si in a similar way: σi strictly dominates si if and only if
σi(si 1 )ui(si 1 , s−i) + σi(si 2 )ui(si 2 , s−i) + · · · σi(sik)ui(sik, s−i) > ui(si, s−i), ∀s−i ∈ S−i.
A rational player i will never play a strategy si iff si is dominated by a (mixed or pure) strategy. Similarly, we can define weak dominance.
Definition 8 A strategy s∗ i weakly dominates si if and only if
ui(s∗ i , s−i) ≥ ui(si, s−i), ∀s−i ∈ S−i
and ui(s∗ i , s−i) > ui(si, s−i)
for some s−i ∈ S−i.
That is, no matter what the other players play, playing s∗ i is at least as good as playing si, and there are some contingencies in which playing s∗ i is strictly better than si. In that case, if rational, i would play si only if he believes that these contingencies will never occur. If he is cautious in the sense that he assigns some positive probability for each contingency, he will not play si. (^2) That is, there is no belief under which he would play si. Can you prove this?
Definition 9 A strategy sdi is a (weakly) dominant strategy for player i if and only if sdi weakly dominates all the other strategies of player i. A strategy sdi is a strictly dominant strategy for player i if and only if sdi strictly dominates all the other strategies of player i.
If i is rational, and has a strictly dominant strategy sdi , then he will not play any other strategy. If he has a weakly dominant strategy and cautious, then he will not play other strategies.
Example:
1 \ 2 work hard shirk hire 2,2 1, don’t hire 0,0 0, In this game, player 1 (firm) has a strictly dominant strategy which is to “hire.” Player 2 has only a weakly dominated strategy. If players are rational, and in addition player 2 is cautious, then we expect player 1 to ”hire”, and player 2 to ”shirk”:^3
1 \ 2 work hard shirk hire 2,2 =⇒ 1, don’t hire 0,0 ⇑ 0,0 ⇑
Definition 10 A strategy profile sd^ = (sd 1 , sd 2 , ....sdN ) is a dominant strategy equilibrium, if and only if sdi is a dominant strategy for each player i.
As an example consider the Prisoner’s Dilemma.
1 \ 2 confess don’t confess confess -5,-5 0,- don’t confess -6,0 -1,-
“Confess” is a strictly dominant strategy for both players, therefore (“confess”, “con- fess”) is a dominant strategy equilibrium. (^3) This is the only outcome, provided that each player is rational and player 2 knows that player 1 is rational. Can you show this?
player bids some bj < b^0 i, player i would get vi − bj under both strategies b^0 i and vi. If the other player bids some bj ≥ vi, player i would get 0 under both strategies b^0 i and vi. But if bj = b^0 i, bidding vi yields vi − bj > 0 , while b^0 i yields only (vi − bj ) / 2. Likewise, if b^0 i < bj < vi, bidding vi yields vi − bj > 0 , while b^0 i yields only 0. Therefore, bidding vi dominates b^0 i. The case b^0 i > vi is similar, except for when b^0 i > bj > vi, bidding vi yields 0 , while b^0 i yields negative payoff vi − bj < 0. Therefore, bidding vi is dominant strategy for each player i.
Exercise 11 Extend this to the n-buyer case.
When it exists, the dominant strategy equilibrium has an obvious attraction. In that case, the rationality of players implies that the dominant strategy equilibrium will be played. However, it does not exist in general. The following game, the Battle of the Sexes, is supposed to represent a timid first date (though there are other games from animal behavior that deserve this title much more). Both the man and the woman want to be together rather than go alone. However, being timid, they do not make a firm date. Each is hoping to find the other either at the opera or the ballet. While the woman prefers the ballet, the man prefers the opera.
Man\Woman opera ballet opera 1,4 0, ballet 0,0 4,
Clearly, no player has a dominant strategy:
Man\Woman opera ballet opera 1,4 ⇐= ⇓ 0, ballet 0,0 ⇑ =⇒ 4,
2.2 Rationalizability or Iterative elimination of strictly domi-
nated strategies
Consider the following Extended Prisoner’s Dilemma game:
1 \ 2 confess don’t confess run away confess -5,-5 0,-6 -5,- don’t confess -6,0 -1,-1 0,- run away -10,-6 -10,0 -10,- In this game, no agent has any dominant strategy, but there exists a dominated strategy: “run away” is strictly dominated by “confess” (both for 1 and 2). Now consider 2’s problem. She knows 1 is “rational,” therefore she can predict that 1 will not choose “run away,” thus she can eliminate “run away” and consider the smaller game
1 \ 2 confess don’t confess run away confess -5,-5 0,-6 -5,- don’t confess -6,0 -1,-1 0,-
where we have eliminated “run away” because it was strictly dominated; the column player reasons that the row player would never choose it. In this smaller game, 2 has a dominant strategy which is to “confess.” That is, if 2 is rational and knows that 1 is rational, she will play “confess.” In the original game “don’t confess” did better against “run away,” thus “confess” was not a dominant strategy. However, 1 playing “run away” cannot be rationalized because it is a dominated strategy. This leads to the Elimination of Strictly Dominated Strategies. What happens if we “Iteratively Eliminate Strictly Dominated” strategies? That is, we eliminate a strictly dominated strategy, and then look for another strictly dominated strategy in the reduced game. We stop when we can no longer find a strictly dominated strategy. Clearly, if it is common knowledge that players are rational, they will play only the strategies that survive this iteratively elimination of strictly dominated strategies. Therefore, we call such strategies rationalizable. Caution: we do eliminate the strategies that are dominated by some mixed strategies! In the above example, the set of rationalizable strategies is once again “confess,” “confess.” At this point you should stop and apply this method to the Cournot duopoly!! (See Gibbons.) Also, make sure that you can generate the rationality as- sumption at each elimination. For instance, in the game above, player 2 knows that
Definition 12 For any player i, a strategy sBRi is a best response to s−i if and only if
ui(sBRi , s−i) ≥ ui(si, s−i), ∀si ∈ Si
This definition is identical to that of a dominant strategy except that it is not for all s−i ∈ S−i but for a specific strategy s−i. If it were true for all s−i, then SBRi would also be a dominant strategy, which is a stronger requirement than being a best response against some strategy s−i.
Definition 13 A strategy profile (sNE 1 , ...sNEN ) is a Nash Equilibrium if and only if sNEi is a best-response to sNE −i = (sNE 1 , ...sNEi− 1 , sNEi+1 , ...sNEN ) for each i. That is, for all i, we have that Ui(sNEi , sN E −i ) ≥ Ui(si, sNE −i ) ∀si ∈ Si.
In other words, no player would have an incentive to deviate, if he knew which strategies the other players play. If a strategy profile is a dominant strategy equilibrium, then it is also a NE, but the reverse is not true. For instance, in the Battle of the Sexes, (O,O) is a NE and B-B is an NE but neither are dominant strategy equilibria. Furthermore, a dominant strategy equilibrium is unique, but as the Battle of the Sexes shows, NE is not unique in general. At this point you should stop, and compute the Nash equilibrium in Cournot Duopoly game!! Why does Nash equilibrium coincide with the rational- izable strategies. In general: Are all rationalizable strategies Nash equilibria? Are all Nash equilibria rationalizable? You should also compute the Nash equilibrium in Cournot oligopoly, Bertrand duopoly and in the commons problem. The definition above covers only the pure strategies. We can define the Nash equi- librium for mixed strategies by changing the pure strategies with the mixed strategies. Again given the mixed strategy of the others, each agent maximizes his expected payoff over his own (mixed) strategies.^4
Example Consider the Battle of the Sexes again where we located two pure strat- egy equilibria. In addition to the pure strategy equilibria, there is a mixed strategy equilibrium. (^4) In terms of beliefs, this correspondes to the requirement that, if i assigns positive probability to the event that j may play a particular pure strategy sj , then sj must be a best response given j’s beliefs.
Man\Woman opera ballet opera 1,4 0, ballet 0,0 4, Let’s write q for the probability that M goes to opera; with probability 1 − q, he goes to ballet. If we write p for the probability that W goes to opera, we can compute her expected utility from this as
U 2 (p; q) = pqu 2 (opera,opera) + p (1 − q) u 2 (ballet,opera)
- (1 − p) qu 2 (opera,ballet) + (1 − p) (1 − q) u 2 (ballet,ballet) = p [qu 2 (opera,opera) + (1 − q) u 2 (ballet,opera)]
- (1 − p) [qu 2 (opera,ballet) + (1 − q) u 2 (ballet,ballet)] = p [q4 + (1 − q) 0] + (1 − p) [0q + 1 (1 − q)] = p[4q] + (1 − p) [1 − q].
Note that the term [4q] multiplied with p is her expected utility from going to opera, and the term multiplied with (1 − p) is her expected utility from going to ballet. U 2 (p; q) is strictly increasing with p if 4 q > 1 − q (i.e., q > 1 / 5 ); it is strictly decreasing with p if 4 q < 1 − q, and is constant if 4 q = 1 − q. In that case, W’s best response is p = 1 of q > 1 / 5 , p = 0 if q < 1 / 5 , and p is any number in [0, 1] if q = 1/ 5. In other words, W would choose opera if her expected utility from opera is higher, ballet if her expected utility from ballet is higher, and can choose any of opera or ballet if she is indifferent between these two. Similarly we compute that q = 1 is best response if p > 4 / 5 ; q = 0 is best response if p < 4 / 5 ; and any q can be best response if p = 4/ 5. We plot the best responses in the following graph.
