Sammary game theory and strategic management

INFORMATION

Imperfect information

• In all previous dynamic games we analyzed, every player observes what other players have played before he

moves. These games are named games of perfect information

• We want to analyze now dynamic games in which there is at least one player who does not observe an action

previously played by another player

• These games are named games of imperfect information

A motivanting example: entry game with simultaneous price decision

• Consider a canonical entry game in which after player 1’s (the entrant) decision whether to enter in a mkt, the

incumbent and the entrant play simultaneously.

• After the entry firms play a simplified Bertrand game in which they can choose to play F or NF.

A motivating example an incorrect representation!

Information sets

• The previous representation neglects an important element of the game

• Firm 1 cannot condition its price strategy to the price chosen by firm 2 because the decision is simultaneous!

• Firm 1 can play either H or L but an action that varies depending on what price the other firm chooses: for instance

firm 1cannot play H if firm 2 ha played H and L if firm 2 plays L.

• We need to specify correctly which information every player has at the time she moves!

• If a player has the same information in two or more different decision nodes, i.e. she cannot distinguish in which

node she is playing, then these nodes belong to the same information set.

• The notion of information set is fundamental in the analysis of games, because a player can choose which action

to play in every information set, not in every decision node.

• A strategy of a player is a specification of an action available, at every information set.

• If two or more decision nodes belong to the same information set we represent that “they are connected” with a

dotted line.

• A player cannot choose two different actions in two nodes that belong to the same information set.

A correct representation

The normal form of the previous game 18

• The Nash equilibria in which player 1 plays Out are sustained by a “threat” that is not credible!

• Notice in fact that “Fight” is a dominated strategy for firm 1.

• Since “No Fight” is a dominant strategy for firm 1, then a rational firm 2 should play NF because it is the only best

response to the dominant strategy NF for firm 1.

• However, if in case of entry both firms play NF, why firm 2 should play Out at the beginning?

Subgame perfect nash equilibrium

• To analyse games of incomplete information we need to introduce the concept of subgames.

• A subgame is a portion of a game that can be logically analysed independently from the rest of the game.

• A subgame (i) starts from a single decision node and contains all the successors; (ii) it can never break an

information set.

A correct representation Subgame perfect nash equilibrium 19

Definition A strategy profile is a Subgame Perfect Nash equilibrium (SPNE) of a game if it induces a Nash

equilibrium in every subgame.

• A game is also a subgame of itself.

• It follows that every SPNE is also a NE but the converse is not true!

• In the motivating initial example the strategy profile such that player 1 plays (IN, NF) and player 2 plays (NF) is the

unique SPNE (but the game has other NE)

Stackelberg game

• Two firms competing a la Cournot (competition in quantities)

• Decisions occur sequentially. Firm 1 (the Leader in the market) decides how much to produce and then (after

having observe what firm 1 has done), firm 2 (the Follower) decides how much to produce.

• Market demand Q=100-p constant marginal costs and equal to 4 for each firm.

• Which are the strategies of the two firms?

• Profits are the same as in the standard Cournot game?

q1

• Suppose firm 1 has produced which is the best response for 2?

• this is firm 2’s reaction curve as in the standard Cournot game.

q1

• Firm 1 can anticipate what firm 2 will produces for any quantity :

Firm 1 maximizes

1. Solve the game and discuss the difference w.r.t a standard Cournot game.

2. Are there other Nash equilibria in this game that are different than the strategy profile that you found?

First mover advantage

• In the SPNE q1=48 q2=24, p=28

• Leader’s profits are higher than follower’s profits and higher of the profits the leader would have obtained in the

simultaneous Cournot game. Leader avails of the first mover advantage.

• In this game there are many Nash equilibria. For instance there is a Nash equilibrium that coincides with the Nash

equilibrium of the simultaneous Cournot game:

• firm 1 produces 32 and firm 2 produces 32 irrespective of the quantity produced by firm 1, e.g. for all q1, q2=32.

• As usual this NE is supported by a “threat” that is not credible.

• When firm 2 observes that firm 1 has produced 48 it is not a best response to produce 32, so the treath of

”producing this quantity irrespective of what the leader has done” is not credible.

Applications: The case of supreme court election

• US Senate nominates judges of the Supreme Court after President’s proposal.

• A candidate is proposed to Senate and in case her/his name does not reach a simple majority of consensus

another candidature is proposed.

• You act as Mister President’s advisor

• For simplicity assume there are three senators (Mr. A, B and C) and four outcomes: three possible candidates and

the outcome of postponing the election after the next presidential election.

• Senators’ preferences are known and represented in the next page.

• The President has to announce to the Senate the list of candidates and the order according to which candidates

will be screened and voted by the Senate.

Senators’ preferences

The alternative “None” is equivalent to postpone the election after next election

• You have to advise Mr. President about the order according to which candidates are listed.

• Suppose that the best candidate for the President is Judge Bork. Cn you suggest an order such that Bork is going

to be elected?

• And in case Kennedy is the best candidate for the President?

• One order to elect Bork is Bork, Ginsberg, Kennedy

• For Kennedy an order is Kennedy, Ginsberg, Bork

Remark: to make a winning candidate is sufficient to put him first in the list! NO! For istance the order Bork, Kennedy,

Ginsberg: Bork is not elected!

Applications: Strategic entry deterrence 20

• Consider a monopolistic market in which only firm A operates

• Firms B and C sequentially decide whether to enter in this market or not.

• Market demand is equal to Q=100-2p

• Marginal cost ar McA=15, McB=12, McC=10

• Firm C is the most efficient firm.

• Suppose after B and C have taken their decision, all firms that entered in the market compete a la Bertrand

• Entry is costly: each firm should pay a sunk cost equal to 200 to enter in the market (not A that is already active in

the mkt)

Applications: ties your hands, most favoured consumer clause

• A Standard Clause allowing a buyer to obtain the best possible price on goods or services from a seller by

requiring it to provide the buyer with the lowest price among all buyers in that market.

• If the seller wants to offer a different buyer a lower price, it must also first offer that price to the buyer with the most

favored customer clause (MFC), also known as a most favored nation clause.

• Why this clause can guarantee your competitors that you will not cut your price in the future?

Best price guarantee

• Why this contract clause may affect competition reducing competitors’ incentive to cut price?

• It represents a credible threat that you will enter in a price war. You make credible your announcement because

your customers guarantee that you will match any price

• Moreover it helps to ne informed about your competitors’ prices!

7. REPEATED GAMES

Repeated games

Up to now we focus our attention on games where there is not repeated interaction over time among players

Nevertheless, in many situations players repeatedly interact among them for a long period.

Examples:

 Firms competition.

 Supply chain relationships.

 Interactions among colleagues in the workplace.

One shot games versus repeated games

• In repeated games, players may condition their strategies to the history of the game (what happened in the past

matters!)

• In repeated games there exist punishments!

 More sophisticated strategies can be played, and more equilibria exist wrt to one-shot games!

A motivating example: inducing cooperation

Which are the NE in pure strategies of this one shot game?

Inducing cooperation

• Suppose players play this game twice.

• After each repetition, players observe what every player has played.

• Players’ payoff is the sum of what they get in each round (discount rate equal to one).

• Is there a SPNE in which the “good” action profile (T,L) is played in some round? 21

• Player 1 (row) plays T in the first

round and plays M in the second if

and only if in the first round (T,L) has

been played, otherwise he plays B.

• Player 2 (column) plays L in the first

round and plays C in the second if

and only if in the first round (T,L) has

been played, otherwise he plays R.

• Let check whether the previous one

is a SPNE: two cases have to be

analyzed Case 1: players played (T,L)

at round 1 Case 2: players did not

play (T,L) at round

A SPNE inducing cooperation

• There are 16 subgames as many as decision nodes of player 1 after the end of round 1.

• For each subgame we have to check that the strategy profile induces a NE in the subgame:

 Case 1: yes! (M,C) is a NE

 Case 2: yes! (B,R) is a NE

• Every player is best responding to the strategy played by the other player in the subgame.

• Finally we have to check that the strategy profile is also a NE of the entire game!

• We have to check that a player has not incentive to deviate from the equilibrium strategy in the first round.

• Suppose player 1 (the argument for player 2 is symmetric!) deviates at round 1

• The best deviation is to play M in the first round:

• The best-deviation strategy gives to player 1 payoff 6 in round 1 and payoff 1 in the second

• If player 1 does not deviate she gets 5 in the first round and 4 in the second: deviating is not profitable!

• Along the equilibrium path, when players observe “cooperation” in round 1, there is a “reward”.

• Out of equilibrium path (but “in equilibrium!”) there is a punishment.

• Both rewards and punishment should be part of an equilibrium strategies.

• Therefore they must be c