Strategy thinking
Incomplete Information
Bayesian Nash Equilibrium and Asymmetric Information
Learning Objectives
Complete versus incomplete information
Bayesian Nash Equilibrium
Games with incomplete information in discrete strategies
Games with incomplete information in continuous strategies
Asymmetric information
Game of Cheap Talk
©Vidya Atal, Montclair State University
Complete Information
All rules of the game fully known by all players and are common knowledge
All strategies, sequence of moves and all pay-offs
Not so common in reality
©Vidya Atal, Montclair State University
Types of Information
Information
Complete information
Incomplete information
Perfect information
Imperfect information
©Vidya Atal, Montclair State University
Symmetric & Asymmetric
4
Incomplete Information in Discrete Strategies
©Vidya Atal, Montclair State University
Incomplete Information and Uncertainty
Uncertainty – random events
Nature’s move, Nature being “player 0”
Treat the two types of Player 1 as two players playing simultaneously
©Vidya Atal, Montclair State University
Bayesian Nash Equilibrium
A Bayesian Nash Equilibrium in a game is
a list of strategies, one for each type of player,
such that no type of player can get a better payoff by switching to some other strategy that is available to her
given the beliefs about the types of players
while all other types of players adhere to the strategies specified for them in the list
©Vidya Atal, Montclair State University
Bayesian Normal Form & Bayesian Nash Equilibrium
Bayesian Nash Equilibrium: (B12B0, D)
©Vidya Atal, Montclair State University
Exercise 1
Consider the following game. Nature selects A with probability ½ and B with probability ½. If Nature selects A, players 1 and 2 interact according to matrix A. If Nature selects B, players interact according to matrix B.
©Vidya Atal, Montclair State University
Suppose that when the players choose their actions, they don’t know which matrix they are playing. Write the normal form matrix that describes this Bayesian game. What is the strategy profile that will be played here?
Exercise 1(a) Answer
This is the case of symmetric incomplete information
Bayesian normal form matrix:
©Vidya Atal, Montclair State University
Equilibrium strategy: (Z, V)
Exercise 1 continued…
©Vidya Atal, Montclair State University
Now suppose that before the players select their actions, Player 1 observes Nature's choice. Player 2 does not observe Nature's choice. Represent this game in the Bayesian normal form. What is the Bayesian Nash Equilibrium in this game?
In this example, is the statement “A player benefits from having more information” true or false?
Exercise 1(b, c) Answer
©Vidya Atal, Montclair State University
Bayesian Nash Equilibrium: (XAYB, W)
Player 1 does NOT benefit from more information
Incomplete Information in Continuous Strategies
©Vidya Atal, Montclair State University
Cournot Game with Incomplete Information
Two firms producing the same good, say bricks
Competing by independently (simultaneously) choosing how much to produce (in thousands), assume all bricks are sold
Price that consumers are willing to pay depends on total number of bricks
Firms’ marginal cost of production:
With probability with probability
Consider 2 types as 2 separate players
©Vidya Atal, Montclair State University
Bayesian Nash Equilibrium in Cournot Duopoly
Profits:
Best response functions:
1 chooses to maximize assuming , fixed
2L chooses to maximize assuming , fixed
2H chooses to maximize assuming , fixed
©Vidya Atal, Montclair State University
Bayesian Nash Equilibrium in Cournot Duopoly
Best response functions:
Bayesian Nash Equilibrium –
©Vidya Atal, Montclair State University
Asymmetric Information
©Vidya Atal, Montclair State University
Asymmetric Information
Incompleteness of information is usually asymmetric
Each player knows her own capabilities and payoffs better than others
Manipulating what others know and believe about you, you can influence equilibrium outcome
Better informed player may want to:
Either conceal information or reveal misleading information (bluff in poker)
Or reveal selected information truthfully (signaling)
Less informed player may want to:
Elicit information or filter truth from falsehood (incentives)
Remain ignorant (credible deniability)
©Vidya Atal, Montclair State University
Revealing Misleading Information
Cheap Talk
©Vidya Atal, Montclair State University
Example: Cheap Talk
Good, Mediocre, or Bad investment
Financial adviser better informed, who may overstate ROI
Adviser’s fee – 2% of $100 investment and 20% of gains
Return – (-50) in B, 1 in M, 55 in G
Reputation cost to adviser for misrepresentation – S if small, L if large; 0<S<L
©Vidya Atal, Montclair State University
Dominated Strategies in Cheap Talk
“Choose I if B” is dominated by “Choose N if B” --- eliminate node a
“Choose I if M” is dominated by “Choose N if M” --- eliminate nodes c, g
“Report B” or “Report M” lead to choosing N
©Vidya Atal, Montclair State University
Best Response Analysis in Cheap Talk
So when S<L<2, “babbling” equilibrium is the only BNE
(Always G, N if G) is a BNE
“babbling” – no information communication
(G only if M or G, I if G) is BNE if S<2.2 and L>2; say S=2, L=3
“partial revelation” – not B
(G if and only if G, I if G) is BNE if S>2.2 and (L+S)>4.2; say S=2.5, L=3
“full revelation” – G only when G
©Vidya Atal, Montclair State University