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I.
EC 630
L12: Games and Imperfect Information
Information and Equilibria in Economic Games
RDC Fall 2010
We now turn to settings in which dominant pure strategies do not exist and/or in which informational problems are explicitly addressed.
A.A good dealof economic analysis and model building implicitly ignores information problems.
i. That is to say, the people modeled are assumed to know as much as can be known about the problem facing them. ii. For example, the setting analyzed may be assumed to be one in which there is complete certainty, and the decision makers simply optimize in a setting where they know all that can be known about their particular problem: e. g. they know their objective function (profit or utility) and their constraints. iii. A similar case occurs when there are random phenomena that affect expected utility levels or payoffs, but where the functions are well understood. The rent-seeking model in the previous lecture was such a case.
iv. If the probability functions are well understood, the persons being modeled simply maximize the "expected" value of their objective functions given their various certain and probabilistic constraints. v. Optimal purchases of lottery tickets, investment in risky processes, rent-seeking, insurance markets etc. are usually modeled in this way.
B.Models that take account of uncertainty are implicitly acknowledging information problems that are relevant for the decisions to be made.
C.In addition to modeling uncertainty, models that take explicit account of imperfect information may also attempt to model learning or changes in information.
i. In such settings, one may be able to predict how new information will affect future choices and so model informational strategies with models that resemble Stackelberg games.
ii. In other cases, one may simply attempt to model how people adapt to different circumstances as they learn a bit more about a particular probablility function, constraint, or their own objectives. iii. What is known, how one comes to know "it" and the manner in which one can learn more about "it" are all matters of interest in models with information problems. wFor example, consider the model of “rational ignorance” developed in the early Public Choice literature by Downs in the 1950s and Tullock in the 1960s.
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If voters or consumers are rationally ignorant, the extent of their knowledge can be “manipulated” by manipulating information costs.
D.In a game setting, knowledge problems can be modeled in several ways.
i. For example, most Cournot models implicitly assume that game participants do not know the strategies adopted by their competitors. In contrast, in Stackelberg games the second mover knows what the first has done. ii. Attempts to prevent the other players from knowing what they will do can also be modeled, as when a player randomly chooses among strategies in a repeated game.
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EC 630
L12: Games and Imperfect Information
RDC Fall 2010
E.his lecture provides an overview of some models of strategic choice in which uncertainty and other inform -tional imperfections play a role. The applications are mostly from the economics literature.
II.
Mixed Strategy Equilibria: "Random" Strategies
A.It bears noting that many "finite" games thatdo nothave a Nash Equilibrium in Pure Strategies.
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For example, many zero sum games lack an equilibrium in pure strategies.
The rock-scissors-paper game also lacks an equilibrium in pure strategies.
Nonetheless, if random play is allowed, there is (nearly) always an equilibrium in mixed strategies.
B.Amixed strategyis a probability function defined over a player's strategy set.
i. The players select among probability functions defined over their strategy sets. wNote that a pure strategy can be regarded as a probability function where the probability of a particular strategy is 1, and the probability of all the other strategies is zero.
ii. It turns out that if the probability of playing one or another strategy is “the” strategy, a 2-strategy game (or other finite strategy game) becomes a game with an continuous strategy set. wThe equilibrium that emerges, however, differs from the Cournot model (in most cases) in that it is driven by consideration of the responses of the opposition.
iii. To calculate payoffs when using mixed strategies, players are normally assumed to maximize theirexpected utility(the expected payoff) associated with their mixed strategy. wFor example, if there are four possible payoffs, U1, U2, U3, and U4and the probability of realizing U1is P1, probability of realizing U2is P2, the probability of realizing U3is P3and or realizing U4is P4.
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E The expected payoff is U = P1U1+ P2U2+P3U3+P4U4
iv. The best reply functions that emerge from this process often have a rather peculiar shape as illustrated below, although the functions will cross at some point.
C.Consider the equilibrium of what Buchanan calls theSamaritan's dilemma(see Buchanan, 1972, Tullock, 1983, or Rasmussen 1994, p. 68). Samaritan's Bum Dilemma Work Loaf
Donor Aid
No aid
3 , 2
1, 1
1, 3
0, 0
i. Note that there is no pure strategy equilibrium to this game. wConsequently, it would not be rational for either player to allow their behavior to be entirely predictable.
ii. To explore the possibility of a mixed strategy equilibrium, suppose that the probability of the Bum working is w and that the probability that the donor will aid the Bum is g.
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L12: Games and Imperfect Information
iv. Differentiating with respect to g and solving allows us to characterize g* w[ 3w + (1-w)(-1) ] - [ (-1)w + (1-w)0 ] = 0
Recall from statistics that thejoint probabilitythatgandwboth happen isgwas long as g and w are chosen independently of one another.)
The best reply function, thus equals 0 for w<.2, 1 for w=2, and our player is indifferent among strategy choices when w=2. [Draw this best reply function.]
iii. he Samaritan's expected payoff from participating in this game is: E wUS= g [ 3w + (1-w)(-1) ] + (1-g) [ (-1)w + (1-w)0 ]
However, if g> 0.5 the Bum's utility falls as w increases, so w* should be set at its lowest value, in which case w*= 0.
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D.if the Donor gives with probability 0.5 and the Bum works with probability 0.2, both are in equilib-Note that rium.
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U= 2gw + w - gw + 3g - 3gw = -2gw + w + 3g
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viii.[Draw these "very strange looking" reaction functions, and note the Nash equilibrium.]
Notice that ifw > 0.2, then utility always increases with g, so g* should be set at its highest value, 1
i. With g=0.5 and w=0.2, both are on their "best reply function" and neither has a reason to change their mixed strategy. ii. (See your diagram of the reaction functions.) iii. No player can achieve a higher expected payoff by changing his or her strategy. iv. Moreover, using any other strategy is likely to be worse,if that alternative strategy is discoveredby the opponent. Page 3
E Thus, we find that dU /dg = 0 for all g if w* = 0.2
or
vii. Differentiating with respect to g, we find that the first order condition for w*:
Again the best reply function has a strange shape. It takes the value 1 for g<0.5, 0 for g>0.5 and the Bum is indifferent among strategies when g=.5.
If the Samaritan sets g< 0.5 , the utility rises as w increases, and w should be set at its highest value, in which case w*=1.
-2g + 1 = 0 which implies that U= 0 only if g = 0.5
vi. A similar calculation for the Bum yields
U= w [ 2g + (1-g)(1) ] + (1-w) [ 3g + 0 (1- g) ]
Note that g is not included in this equation!
And, ifw < 0.2then utility always falls with g, so g" should be set at its lowest value, 0.
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v. This suggests that g does not have a simple "interior solution." w3w - 1 +w ] + [ w ] = 0 => [ => 5w = 1Solving for w, using the first order condition
EC 630
e. g. if the Samaritan sets g = .5 , then any value ofwwill generate the same utility for the Bum.
EC 630
L12: Games and Imperfect Information
RDC Fall 2010
E.Oneeculiar propertof mostmixed strategy equilibriais that at the equilibrium each player isindifferen among other (non-equilibiurm) probabilities given the other's behavior, butprobabilityno other combination generate an equilibrium.
i.Giventhe behavior of other player(s), each player is indifferent among all strategies because they all generate the same expected payoff (expected utility) w(Recall the first order conditions above.)
ii. This equilibrium is also a bit demanding in thatit assumes that all players understand the entire game, not simply their part of it. wThis requires them to "know" the other player's strategies, payoffs, and method for making decisions in a setting of uncertainty.
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At a Nash Equilibria in pure strategies, players do not have to understand the behavior of other players to realize that they should not change their strategies.
iii. In other words, achieving a "Mixed Equilibrium Strategy" requires a very sophisticatedconsciously game theoretic mode of thinkingby the players (as in a Stackelberg model, rather than a Cournot model). wAll players have to understand the nature of the game and anticipate the rational responses of other players.
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To do this, they must know a lot about the other players.
[Note that if one of the players mistakenly plays a pure strategy, no one is made worse off, but if this pure strategy players is discovered the other players will want to change their strategies. Why? (Hint: think of the paper, rock scissors game.)]
iv. Of course, the same mode of reasoning applied to a game with a deterministic equilibrium implies that players will go directly to the Nash equilibrium. wFortunately, in many cases the Nash equilibriummay emergefrom a series of "moves" even if the players behave "myopicly" rather than fully rationally.
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(Myopic players optimize only one move at a time, rather than taking account of all possible future moves of opponents.)
(See diagram of myopic adjustment in, for example, the lottery game.)
F.Our results demonstrate thatequilibrium mixed strategies can be computed by solving for a probability function that makes the other player(s) indifferent among their strategy choices.
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[Student Puzzle: fine the mixed strategy equilibria for a "zero sum" game; and for the Paper, Stone, Scissors game played by children. How do your choices of payoffs affect the "mixing probabilities?]
[Student puzzle: It can be said that a mixed strategy equilibrium depends on "random play" being "perfectly predictable" How is this possible?]
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EC 630
L12: Games and Imperfect Information
III. A Quick Review of Rational Choices in a Stochastic Environment
RDC Fall 2010
A.The mixed strategy equilibrium is only one of many settings in which choices must be made in an environment in which particular outcomes are not known at the moment choices have to be made.
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The simplest cases are coin tosses, rolling dice, and lotteries.
The rent-seeking contests of the previous lecture are also examples.
More complex cases arise whenever a long term effect has to be estimated in some way, as with future interest rates, weather, or political contests.
B. A Review of Probabilities and Probability Functions.One way to characterize a stochastic environment is with aprobability functionmapped over the "event space."
i. A probability function has several important properties. ii. Everypossible outcomeis assigned a number between 0 and 1. wIf the possible outcomes are X1, X2, X3... XNthen aprobability functiondefined over those possibilities looks like Pi= f(Xi) , for i = 1, 2, .... N.
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Probabilities cannot, by definition, be a negative number.
Every impossible outcome is assigned the value 0.
iii. The sum of the probabilities, by definition, has to add up to 1.
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N i=1Pi)= 1
C.The "expected value" of a random event such as X above is a technical statistical term that means "average" outcome.
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It is widely used in rational choice models and in applied work by statisticians.
e N The expected value of X can be calculated as: X =i=1PiXi
For example in rolling a dice, Pi = 1/6 for i = 1, 2 ... 6 and X1= 1, X2= 2, .... X6= 6
e N X =i=1PiXi= (1/6)(1) + (1/6)(2) + (1/6)(3) + (1/6)(4) + (1/6)(5) + (1/6) 6 =3.5
(As this illustration illustrates, sometimes the expected value is actually impossible. However, if one calculated the average of a large number of dice throws (say 25 or more) the result would be number quite close to 3.5. )
D. Expected Utility. For many purposes, in rational choice theory only the "ordinal" properties of utility functions matter.
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That is to say, the fact that U(A) = 2 * U(B) only means that A is preferred to B, not that A is twice as desirable as B.
i. However,to calculate expected utilityin the conventional ways, requires utility functions for which such algebra has meaning. wOnly a subset of possible utility functions can be used to calculate "expected utilities."
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L12: Games and Imperfect Information
hese are called Von Neuman Morgenstern utility functions.
RDC Fall 2010
ii.Von-Neuman Morgensternutility functions are bounded and continuous. wVon-Neuman Morgenstern utility functions are "unique" up to a linear transformation.
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Consequently, they are regarded by many economists and decision theorists to representquais-cardinal utility (in addition to ordinal utility).
e V-N Morgenstern Utility functions have the property that if U (A) > Ue(B), then A is preferred to B, e e where U (A) is the "expected utility of stochastic setting A" and U (B) is the expected utility of stochastic setting B.
iii. Theexpected utilityassociated with such a stochastic setting is calculated as the expected value of utility: N e E(U(X)) = U =PiU(Xi) i = 1 w where thevalue of the possible outcomes is now measured in utility terms.
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e E(U(V)) is morecommonly written as U (X)
e e iv. It is important to remember that U (X) isnotgenerally equal to U(X ). e e wU (X) isthe same asU(X )only if U is a linearfunction of V.
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In such cases, the individual is said to berisk neutral.
v. DEF: An individual is said to berisk averseif the expected utility of some gamble or risk is less than the utility generated at the expected value (mean) of the variable being evaluated. wArisk neutralindividual is a person for whom the expected utility of a gamble (risky situation) and utility of the expected (mean) outcome are the same.
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A risk preferring individual is one for whom the expected utility of a gamble is greater than the utility of the expected (mean) outcome.
vi. The more risk averse a person is, the greater is the amount that he or she is willing to pay to avoid a risk.
E.A Von Neumann Morgenstern Utility function can be constructed as follows.
i. Assume that Xmin and Xmax are the best and worse payoffs of the game. wRecall, in a finite game, all the payoffs are finite, so Xmin and Xmax exist, and are often very easy to calculate.
ii. The utility of Xi where Xmin<Xi<Xmax can be calculated as a "convex combination" of the arbitrary values U(Xmin) and U(Xmax) where U(Xmin) < U(Xmax). wRecall that a convex combination of X and Y can be written as Z = (1-a) X + (a) Y where 0<a<1.
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Note that "a" can thus be interpreted as the probability of X, and Z as the expected value of getting X with probability "a" and Y with probability "1-a."
iii. Note also that this "probability" can be varied until the individual of interest is indifferent between the gamble and Xi. wThe utility value of Xi is then defined to be:
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U(Xi) = p U(Xmin) + (1-p) U(Xmax). Page 6
EC 630
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L12: Games and Imperfect Information
RDC Fall 2010
(That is to say, assign utility numbers to events (Xi) by varying P until the person of interest is indifferent between the event (Xi) with certainty and the gamble over Xmin and Xmax defined by P.)
iv. The Von-Neumann Morgenstern utility index was worked out in their very important game theory textThe Theory of Games and Economic Behavior, 1944. w(Student puzzle: Why does the difference between ordinal and cardinal utility matter?)
IV.
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(Student relief: for most of the rest of this course, we require only "ordinal" utility functions.)
Practice Problems
A.-1 and 1 in various orders.Create a zero sum game 2x2 game using the payoffs of
i. Does your game have a Nash equilibrium in deterministic strategies? ii. Does your game have any Pareto optimal outcomes? iii. If there is a deterministic equilibrium, change your payoffs to eliminate this equilibrium.
iv. Find the mixed strategy equilibrium for your game, and explain why it is an equilibrium.
B.Repeat "A" with a two person three strategy game.
C.Repeat "A" again using zero sum payoffs from -3 to +3.
D.Create a two stage game in which in stage one, players may send signals (creditable commitments) to each other about whether they will cooperate or not in the PD game that follows.
V.
i. Assume that creditable commitments are possible, but that creating them is costly. ii. Find the equilibrium strategies in stages 1 and 2 of this game. iii. Can communication in this type of setting solve the PD game?
iv. How costly does a creditable commitment have to be to make a commitment to cooperate a poor strategy? v. How might a game be designed to make "truth telling" an optimal strategy? (Discuss a couple of examples from ordinary markets regarding quality or price.)
Information Sets, Ignorance, and Game Theory
A.In the last section, we also mentioned that games in extended form can be used to represent simultaneous play, by making assumptions about what players know as the game unfolds.
i. Specifically, if player 2 goes second, but does not know what the first player has done, it is as if player 2 is in a simultaneous game. ii. (Ignorance of the other player's strategy choice is the most important consequence of simultaneous play in most parlor games.)
B.Formally, what we have done is to characterize theinformation setsof participants in the game.
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EC 630
L12: Games and Imperfect Information
RDC Fall 2010
i. Rasmusen, p. 40, defines aninformation setPlayer i's information setas follows: iat any particular point of the game is the set of different nodes in the game tree that "he" knows might be the actual node, but between which he cannot distinguish by direct observation. ii. We represented this last time with dotted lines surrounding the nodes of the second player after the first had chosen to defect or cooperate in the PD game.
C.A related concept is a player'sinformation partition.
i. Again Rasmusen, p. 42, provides a useful definition: a player's information partition is a collection of his information sets such that: (1) each path is represented by one node in a single information set in the parti-tion and (2) the predecessors of all nodes in a single information set are in one information set. ii. The information partition represents the different positions that the player knows he will be able to distin-guish from each other at different stages of the game, thereby dividing up the set of all possible nodes into subsets called information sets. iii. If a player knows exactly what has happened at every stage in the game to the point of interest, the informa-tion partition is simply the set of all nodes that can be reached. E. g. each information set is a "singleton" consisting of the actual node itself.
iv. The finer the informational partition, (smaller the information sets) the better is the player's information. (The player will better be able to distinguish between nodes in the game if the information sets are smaller.) v. Information is said to beCommon Knowledgeif it is known to all players, if each player knows that all the players know it, if each player knows that all the players know that all the players know it, and so forthad infinitum.
VI.
Bayes' Law, Signaling, and Learning Functions
A.Learning in most games and in most economic and statistical models is represented as changes in a probability function that characterizedexpectedoutcomes or possibilities.
i. For example, a consumer might have an general idea about the range of prices that he or she will have to pay for gasoline in Alaska (this would be called apriorprobability distribution). ii. Once you have actually purchased gasoline there, or done a bit of research, the probability function that you assign to Alaskan gasoline prices will change to reflect the new information that you have. You have learned something new about that distribution. The new distribution is called aposteriorprobability function. iii. One can model this learning process in a general way as F1(x) = p( x | Fo, S ) where F1is the conditional (posterior) density function of x given prior probability function Fo, and S is some signal, observation, message, or event that causes the individual to change (or update) "his" priors. F1(x) becomes the new prior. So the next updated posterior would be F2(x) = p(x| F1, S). wGenerally as one becomes better informed, the probability function describing beliefs becomes "tighter" and converges toward the real value of x. That is to say, the variance of F generally falls with learning.
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(On the other hand, occasionally, one learns jsut how ignorant one really is, in which case the variance of this probability function may increase.)
iv.Diffuse priorsWith diffuse priors, one regards everyare often the initial starting point of this kind of analysis. possibility to be equally likely. That is to say p(xi), where i = 1 ..... N , is 1/N for all i. v. Bayes law is derived from statistical definitions of joint probabilities: Page 8
EC 630
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L12: Games and Imperfect Information
he joint probability of E and H being true is P(E, H) = P(H|E) p(E) = P(E|H) p(H)
Using the last equality we can solve for P(H|E) as: P(H|E) = P(E|H) p(H) /p(E)
RDC Fall 2010
vi. This isBayes' lawthat relates the posterior probability P ( H|E) to the prior probability that H is true, P(H), the conditional probability that E will be observed if H is true, P(E|H), and the probability that E will be observed in any case, P(E). (Varian, p. 192) wBayes' law is important because it shows how a rational individualshould(rationally) update his priors as new information is received.
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Since Bayes' law follows directly from statistical notions of joint probability, the law applies equally to internally consistent subjective and consistant objective probabilities.
B.In a signaling game individuals send messages to each other with the intent of changing the other player's expectations and thereby their behavior. In most models of signalling behavior, the signals / messages are the only control variable, or one of two control variables. (Illustration as a Nash Game: see class notes)
i. The problem ofcheap talkin such games addresses why should anyone believe a signal? ii. Suppose there is a two stage game. In the first stage a person can signal his "intention," in the second which is a traditional PD game. (Either player may signal I "will " or "will not" cooperate in the PD game that follows.)
iii. If there is no penalty for lying there is no reason to believe the signals in the first round. w(Puzzle: Why then do experiments where people can talk yield more cooperation than those where players do not?)
C.A related game concernsscreeningin a setting wherepeople into different categories, such as productivity, the "firm owner" knows less about the productivity of his or her potential employees than they do themselves.
i. Here all potential employees attempt to signal the employer about their productivity, but in many cases they will have an incentive to exaggerate their productivity (as in team production settings). ii. The challenge for a firm owner or manager is to devise a signaling game that actually produces useful infor-mation, or, alternatively, self selection into separate pools of productivitys. iii. For example, commissions for sales can be paid, rather than salaries. This induces persons who expect to be good salesmen to accept the commission and induces others to seek other positions in the firm or to shift to other firms.
VII. Homework Problems (Optional)
A.-1 and 1 in various orders.Create a zero sum game 2x2 game using the payoffs of
i. Does your game have a Nash equilibrium in deterministic strategies? ii. Does your game have any Pareto optimal outcomes?
iii. If there is a deterministic equilibrium, change your payoffs to eliminate this equilibrium. iv. Find the mixed strategy equilibrium for your game, and explain why it is an equilibrium.
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L12: Games and Imperfect Information
RDC Fall 2010
B.model in a PD game where one prisoner can attempt to persuade the other thaCreate a Stackelberg signaling he will cooperate. Use an abstract learning function and assume signals are costly, and that both players maximize expected utility.
C.Analyze and Discuss.Is a costly signal necessarily a creditable signal?
D.How might a game be designed to make "truth telling" an optimal strategy? (Discuss a couple of examples from ordinary markets regarding quality or price.) Analyze and Discuss.
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