ECO290E: Game Theory

Lecture 12

Static Games of Incomplete Information

Review of Lecture 11

In the repeated Bertrand games, the following “trigger” strategies achieve collusion if δ≥1/2.

• Each firm charges a monopoly price until someone undercuts the price, and after such deviation she will set a price equal to the marginal cost c, i.e., get into a price war.

t t+1 t+2 …

collusion π π π …

deviation 2π 0 0 …

Finite RepetitionsQ: If the Bertrand games are played only finitely, i.e.,

ends in period T, then collusion can be sustained?A: NO!• In the last period (t=T), no firm has an incentive to

collude since there is no future play. The only possible outcome is a stage game NE.

• In the second to the last period (t=T-1), no firm has an incentive to collude since the future play will be a price war no matter how each firm plays in period T-1.

• By backward induction, firms end up getting into price wars in every period.

No collusion is possible!

Finitely Repeated Games

• If the stage game G has a unique NE, then for any T, the finitely repeated game G(T) has a unique SPNE: the NE of G is played in every stage irrespective of the history.

• If the stage game G has multiple NE, then for any T, any sequence of those equilibrium profiles can be supported as the outcome of a SPNE. Moreover, non-NE strategy profiles can be sustained as a SPNE in this case.

Games of Incomplete Information

• In a game of incomplete information, at least one player is uncertain about what other players know, i.e., some of the players possess private information, at the beginning of the game.

• For example, a firm may not know the cost of the rival firm, a bidder does not usually know her competitors’ valuations in an auction.

• Following Harsanyi (1967), we can translate any game of incomplete information into a Bayesian game in which a NE is naturally extended as a Bayesian Nash equilibrium.

Cournot Game with Unknown Cost

• Firm 1’s marginal cost is constant (c), while firm 2’s marginal cost takes either high (h) with probability θ or low (l) with probability 1-θ.

• Firm 1’s strategy is a quantity choice, but firm 2’s strategy is a complete action plan, i.e., she must specify her quantity choice in each possible marginal cost.

• Assume each firm tries to maximize an expected profit.

Calculation

• It is important to consider the types of player 2 as separate players.

• Equilibrium strategies can be derived by the following maximization problems:

)(}))(({)(max

)(}))(({)(max

}))((){1(

}))(({max

2212)(

2212)(

121

1211

2

2

1

lqllqqbal

hqhhqqbah

qclqqba

qchqqba

lq

hq

q

Solution

• Notice that firm 2 will produce more (/less) than she would in the complete information case with high (/low) cost, since firm 1 does not take the best response to firm 2’s actual quantity but maximizes his expected profit.

)(63

2)(

)(6

1

3

2)(

3

)1(2

*2

*2

*1

lhbb

clalq

lhbb

chahq

b

lhcaq

Static Bayesian Games

1. Nature draws a type vector t, according to a prior probability distribution p(t).

2. Nature reveals i’s type to player i, but not to any other player.

3. The players simultaneously choose actions.

4. Payoffs are received.

Remarks

• A belief about other players’ types is a conditional probability distribution of other players’ types given the player’s knowledge of her own type.

• A (pure) strategy for a player is a complete action plan, which specifies her action for each of her possible type.

• Bayes’ rule:

)Pr(

),Pr()|Pr(

B

BABA

Bayesian Nash Equilibrium

• A strategy profile s* is a Bayesian NE if

which is equivalent to

t tiiiiiiiiiii

i

ttstsuttstsu

si

));(),(());(),((

)(***

i

i

tiiiiiiii

tiiiiiiii

ii

ttttstsu

ttttstsu

tsi

)|Pr());(),((

)|Pr());(),((

)(

*

**

Calculation

• Maximizing RHS is identical to maximizing inside the brackets for all possible i’s type.

i it tiiiii

tiiiii

tiii

tttttsu

tttttsu

tttsuuE

)Pr()]|Pr());(([

)Pr()|Pr());((

)Pr());((][