Incomplete Information Econ 400 University of Notre Dame Econ 400 (ND) Incomplete Information 1 / 25
Games of Incomplete Information In game theory, there are two sources of uncertainty related to information: Econ 400 (ND) Incomplete Information 2 / 25
Games of Incomplete Information In game theory, there are two sources of uncertainty related to information: Uncertainty about the preferences or capabilities of an opponent (Incomplete Information) Econ 400 (ND) Incomplete Information 2 / 25
Games of Incomplete Information In game theory, there are two sources of uncertainty related to information: Uncertainty about the preferences or capabilities of an opponent (Incomplete Information) Uncertainty about the previous actions of other players (Imperfect Information) Econ 400 (ND) Incomplete Information 2 / 25
Games of Incomplete Information In game theory, there are two sources of uncertainty related to information: Uncertainty about the preferences or capabilities of an opponent (Incomplete Information) Uncertainty about the previous actions of other players (Imperfect Information) The first kind of uncertainty can actually be converted into the second, actually. Econ 400 (ND) Incomplete Information 2 / 25
A Simple Game with Incomplete Information There are two players with two strategies each, S or C . However, the payoff matrix is c S C r S 1,1 -1,x c C x r , − 1 0,0 where x r is known only to player r , and x c is known only to player c . The value x r is player r ’s type , and the value x c is player c ’s type . Econ 400 (ND) Incomplete Information 3 / 25
A Simple Game with Incomplete Information: Types and Beliefs The players’ types follow this distribution: With probability p , x r = 0, and with probability 1 − p , x r = 2 With probability p , x c = 0, and with probability 1 − p , x c = 2 Econ 400 (ND) Incomplete Information 4 / 25
A Simple Game with Incomplete Information: Types and Beliefs The players’ types follow this distribution: With probability p , x r = 0, and with probability 1 − p , x r = 2 With probability p , x c = 0, and with probability 1 − p , x c = 2 But because the players’ types are random and unknown to each other, they are uncertain about “who” their opponent is. However, they have the same beliefs about frequencies of opponent types, like knowing the composition of the deck of cards from which hands are dealt. Econ 400 (ND) Incomplete Information 4 / 25
A Simple Game with Incomplete Information: Payoffs There are actually four games that might be going on: c S C r S 1,1 -1,0 C 0,-1 0,0 Econ 400 (ND) Incomplete Information 5 / 25
A Simple Game with Incomplete Information: Payoffs There are actually four games that might be going on: c c S C S C r S 1,1 -1,0 r S 1,1 -1,2 C 0,-1 0,0 C 2,-1 0,0 Econ 400 (ND) Incomplete Information 5 / 25
A Simple Game with Incomplete Information: Payoffs There are actually four games that might be going on: c c S C S C r S 1,1 -1,0 r S 1,1 -1,2 C 0,-1 0,0 C 2,-1 0,0 c S C r S 1,1 -1,2 C 0,-1 0,0 Econ 400 (ND) Incomplete Information 5 / 25
A Simple Game with Incomplete Information: Payoffs There are actually four games that might be going on: c c S C S C r S 1,1 -1,0 r S 1,1 -1,2 C 0,-1 0,0 C 2,-1 0,0 c c S C S C r S 1,1 -1,2 r S 1,1 -1,0 C 0,-1 0,0 C 2,-1 0,0 The players just aren’t sure which one they’re actually in. Econ 400 (ND) Incomplete Information 5 / 25
A Simple Game with Incomplete Information: Strategies What are strategies? A strategy for the row player is a rule saying what the row player should do — S or C — for each type x r = 0 or x r = 2. There are four possibilities: (0,2) → ( S , S ) (0,2) → ( S , C ) (0,2) → ( C , S ) (0,2) → ( C , C ) Econ 400 (ND) Incomplete Information 6 / 25
A Simple Game with Incomplete Information: Strategies What are strategies? A strategy for the row player is a rule saying what the row player should do — S or C — for each type x r = 0 or x r = 2. There are four possibilities: (0,2) → ( S , S ) (0,2) → ( S , C ) (0,2) → ( C , S ) (0,2) → ( C , C ) Similarly, the column player has four potential strategies: (0,2) → ( S , S ) (0,2) → ( S , C ) (0,2) → ( C , S ) (0,2) → ( C , C ) Econ 400 (ND) Incomplete Information 6 / 25
Expected Payoffs Note that if the column player’s strategy is, say, (0 , 2) → ( C , S ), then the probability that column uses C is p , and the probability column uses S is 1 − p . This means that we can compute expected payoffs by using the rule that corresponds to each type. Econ 400 (ND) Incomplete Information 7 / 25
Expected Payoffs Note that if the column player’s strategy is, say, (0 , 2) → ( C , S ), then the probability that column uses C is p , and the probability column uses S is 1 − p . This means that we can compute expected payoffs by using the rule that corresponds to each type. For example, if column uses (0 , 2) → ( C , S ), then row’s payoff is pu row ( σ row , C , t row , 0) + (1 − p ) u row ( σ row , S , t row , 2) Econ 400 (ND) Incomplete Information 7 / 25
Expected Payoffs Note that if the column player’s strategy is, say, (0 , 2) → ( C , S ), then the probability that column uses C is p , and the probability column uses S is 1 − p . This means that we can compute expected payoffs by using the rule that corresponds to each type. For example, if column uses (0 , 2) → ( C , S ), then row’s payoff is pu row ( σ row , C , t row , 0) + (1 − p ) u row ( σ row , S , t row , 2) or if column uses (0 , 2) → ( C , C ), then row’s payoff is pu row ( σ row , C , t row , 0) + (1 − p ) u row ( σ row , C , t row , 2) Econ 400 (ND) Incomplete Information 7 / 25
Equilibrium A set of strategies for each player-type is a Bayesian Nash equilibrium if no player-type can deviate and get a higher expected payoff. Or, if we make a table, Type Strategy t 11 s 11 . . . . . . t 22 s 21 . . . . . . t NK s NK So player i ’s k -th type, t ik , row ik is assigned a strategy s ik . A table like the one above is an equilibrium if no player-type t ik can choose a strategy s ′ that gives a strictly higher payoff than s ik . Econ 400 (ND) Incomplete Information 8 / 25
Equilibrium 1: Always confess Suppose Type Strategy x r = 0 C x r = 2 C x c = 0 C x c = 2 C so everyone confesses no matter what. Is this a Bayesian Nash equilibrium? Econ 400 (ND) Incomplete Information 9 / 25
Equilibrium 1: Always confess We check that no player-type wants to deviate (for each row in the table, playing as suggested is better than switching to something else): Does the 0-type row player want to deviate? Econ 400 (ND) Incomplete Information 10 / 25
Equilibrium 1: Always confess We check that no player-type wants to deviate (for each row in the table, playing as suggested is better than switching to something else): Does the 0-type row player want to deviate? The expected payoff from confessing (assuming all other types do what it says in the table) is p (0) + (1 − p )(0) = 0 The expected payoff from deviating and remaining silent (assuming all the other types do as the table says) is p ( − 1) + (1 − p )( − 1) = − 1 Econ 400 (ND) Incomplete Information 10 / 25
Equilibrium 1: Always confess We check that no player-type wants to deviate (for each row in the table, playing as suggested is better than switching to something else): Does the 0-type row player want to deviate? The expected payoff from confessing (assuming all other types do what it says in the table) is p (0) + (1 − p )(0) = 0 The expected payoff from deviating and remaining silent (assuming all the other types do as the table says) is p ( − 1) + (1 − p )( − 1) = − 1 So the 0-type row player doesn’t want to deviate. Econ 400 (ND) Incomplete Information 10 / 25
Equilibrium 1: Always confess Does the 2-type row player want to deviate? The expected payoff from confessing (assuming all the other types do as the table says) is pu r ( C , C , 2) + (1 − p ) u r ( C , C , 2) = 0 The expected payoff from deviating to remaining silent (assuming all the other types do as the table says) is p ( − 1) + (1 − p )( − 1) = − 1 Econ 400 (ND) Incomplete Information 11 / 25
Equilibrium 1: Always confess Does the 2-type row player want to deviate? The expected payoff from confessing (assuming all the other types do as the table says) is pu r ( C , C , 2) + (1 − p ) u r ( C , C , 2) = 0 The expected payoff from deviating to remaining silent (assuming all the other types do as the table says) is p ( − 1) + (1 − p )( − 1) = − 1 So the 2-type row player doesn’t want to deviate either. Econ 400 (ND) Incomplete Information 11 / 25
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