11 & 12 - Dynamic Games of Incomplete Information

Question 1

How do you deal with noncredible threats in dynamic games?

Answer 1

• perfect information: SPNE eliminates all noncredible threats
• imperfect information: some non-credible threats may remain even after imposing subgame perfection -> further refinement: Perfect Bayesian equilibrium (PBE)

Question 2

How are Dynamic Games of Incomplete Information represented?

Answer 2

Dynamic games of incomplete information are represented in extensive form by adding Nature as a player who moves first

Question 3

How are Dynamic games of incomplete information usually analyzed?

Answer 3

using Perfect Bayesian Equilibrium (or some closely related equilibrium concept)

Question 4

What are the four requirements for Perfect Bayesian Equilibrium?

Answer 4

1) Beliefs: At each information set, the player with the move must have a belief about which node in the information set has been reached by the play of the game.
2) Sequential Rationality: Given their beliefs, the players’ strategies must be sequentially rational
3) Consistent Beliefs: At information sets on the equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategies.
4) Off-the path: At information sets off the equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategies where possible.

Question 5

What is a belief in PBE requirement 1?
(“At each information set, the player with the move must have a belief about which node in the information set has been reached by the play of the game.”)

Answer 5

• For a nonsingleton information set, a belief is a probability distribution over the nodes in the information set.
• For a singleton information set, the belief puts probability 1 on that single decision node.

More Formally: system of beliefs μ in an extensive-form game is a specification of a probability μ(x) ∈ [0, 1] for each decision node x such that Σ μ(x) = 1 for all information sets H.

Question 6

What does “sequentially rational” mean in PBE requirement 2?
(“Given their beliefs, the players’ strategies must be sequentially rational”)

Answer 6

At each information set, the action taken by the player with the move (and the player’s subsequent strategy) must be optimal given the player’s belief at that information set and the other players’ subsequent strategies.

Question 7

What is a subsequent strategy in PBE?

Answer 7

A subsequent strategy is a complete plan of action covering every contingency that might arise after a given information set has been reached.

Question 8

What does it mean to say an information H set is on the equilibrium path?

Answer 8

For a given equilibrium strategy profile σ, an information set H is
• on the equilibrium path if H will be reached with strictly positive probability,
• off the equilibrium path if H is certain not to be reached.

Question 9

How can you calculate players’ beliefs at information sets on the equilibrium path?

Answer 9

Given a strategy profile σ, a decision node x is reached with probability Pr[x | σ].

If information set H has been reached, a player believes to be in decision node x ∈ H with probability Pr[x | H, σ]

Question 10

What is the typical setup and timing for a signalling game?

Answer 10

1) Nature draws a type t for the Sender from a finite type space T according to a probability distribution p(t),
where p(t) > 0 for all t ∈ T and Σ p(t) = 1.
2) The Sender observes t and then chooses a message m from a finite message space M.
3) The Receiver observes m (but not t) and chooses an action a from a finite action space A.
4) Payoffs uₛ(m, a; t) and uᵣ(m, a; t) are received.

Question 11

What are pooling and separating strategies in a signalling game?

Answer 11

• pooling if all types send the same message
• separating if each type sends a different message
• partially pooling or semi-separating if some (but not all) types send the same message [if more than two types]

Note: A mixed strategy for the Sender can be hybrid: e.g., t₁ sends m₁ and t₂ randomizes between m₁ and m₂

Question 12

To find the PBE in signalling games, how can you interpret requirement 1 (beliefs)?

Answer 12

After observing any message m ∈ M, the Receiver must have a belief about which types could have sent m. Denote this belief by the probability distribution μ(t | m), where μ(t | m) ≥ 0 for each t ∈ T, and
Σ μ(t | m) = 1.

Question 13

To find the PBE in signalling games, how can you interpret requirement 2 (Sequential Rationality)?

Answer 13

At each information set, a player’s action must be optimal given her belief and the other players’ subsequent strategies.
• For the Sender, beliefs are trivial.
• For the Receiver, there are no subsequent strategies.
-> thus both must maximize their utility given their beliefs/subsequent strategies.

Question 14

To find the PBE in signalling games, how can you interpret requirement 3 (consistent beliefs)?

Answer 14

Requirement 3: At information sets on the equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategies.

Question 15

To find the PBE in signalling games, how can you interpret requirement 4 (strategy off equilibrium path)?

Answer 15

Requirement 4 is without effect: At information sets off the equilibrium path, the Receiver’s belief cannot be derived from the Sender’s strategy.
-> For messages m that are never sent, any belief μ(t | m) is consistent with m*.

Question 16

What is a Perfect Bayesian Equilibrium in Signaling Games?

Answer 16

A pure-strategy perfect Bayesian equilibrium in a signaling game is a strategy profile (m, a) and a system of beliefs μ(t | m) satisfying Signaling Requirements 1, 2, and 3.
• Sender plays pooling strategy -> pooling equilibrium
• Sender plays separating strategy -> separating equilibrium

Question 17

How do you find the pure-strategy Nash Equilibria in any extensive-form game?

Answer 17

write the game down in normal-form and find the pure-strategy nash equilibria through best-response

Question 18

How do you find the pure-strategy Perfect Bayesian Nash Equilibria in any extensive-form game?

Answer 18

1) All PBE are also pure-strategy N.E., thus first find the pure strategy N.E. as potential candidates
2) Check whether for player 1 the strategy is sequentially rational, given player 2’s strategy
3) Check for player 2:
- Any beliefs about p, 1-p from player 1’s strategy?
- Resulting from this, which values for p are consistent with the strategy profile?
- check if the suggested strategy yields the largest payoff given any p? What are the conditions for p?

Question 19

How do you write down a pure-strategy Perfect Bayesian Equilibrium?

Answer 19

{ (σ₁, σ₂), (p, 1-p) } ₚ∈[ₓ,ᵧ]

Question 20

How to find a mixed strategy PBE?

Answer 20

1) check cases with one player playing pure, would the other randomize?
2) check in information set which believe would lead to mixed strategy (p when indifferent between two strategies)
3) Check in which cases this p would be consistent and if these cases are possible
4) check the strategy profile of mixing required for the other player to play that case

Question 21

If you have identified all proper subgames of a game, what does sequential rationality imply for behavior in a PBE in these subgames?

Answer 21

If no information sets contained: beliefs are trivial. Solving with sequential rationality only.

Question 22

How to solve for pure strategy PBE in signaling games?

Answer 22

First, check candidates for equilibria: are there strictly dominated strategies for the sender? if not, check all pooling & separating equilibria step by step.

1) What are the receiver’s beliefs about μₓ?
2) What is the rational strategy for the receiver on the equilibrium path (expected payoffs based on beliefs)?
3) Would the sender have an incentive to deviate from the equilbrium path, given this strategy?
4) What must be the receiver’s beliefs off the equilibrium path to play either strategy?
5) Would the sender have an incentive to deviate if the receiver changes her strategy off the equilibrium path?

Question 23

What is the intuitive criterion?

Answer 23

if a deviation from the equilibrium does not pay off for one type (even under the most optimistic beliefs for this type), then one must believe it was the other type