Game Theory and Models of Oligopoly
- Main Source: text Ch. 13
- Some interesting additional sources:
Dixit and Nalebuff (2008) The Art of Strategy (an intro with applications:
we will use some of their examples)
K. Binmore (2007) Game Theory: a Very Short Introduction.
- Behavioral assumptions in the Cournot, Bertrand and Stackelberg models
are quite simple.
- Too simple?
- Are the outcomes realistic?
- Why don’t the oligopolists cooperate to maximize joint profits?
- Game theory offers more flexible ways of thinking about strategic
interaction among firms.
- Wider application: a way of thinking about strategic action between
any decision-makers.
- Including non-economic decisions.
- Many game theory applications outside of economics:
politics, sports, sociology, biology (evolution) etc.
- What is a game?
“A situation of strategic interdependence: the outcome of your choices
depends upon the choices of one or more other persons acting
purposely.” (Dixit and Nalebuff, 2008)
- Elements of a game theoretic model:
Players: decision-makers
Strategies (“moves”, “actions”): possible decisions of the players.
Payoffs: the gains/losses for each combination of strategies chosen by
the players.
Decision rule: rule(s) identifying how agents choose strategies.
- choose the strategy that is best for you (egoist)
- how to break ties? e.g. if two strategies are
equally good for you.
- choice when opponents’ choices are uncertain
e.g. should you seek to limit possible losses
(text: ‘minimax’)? Or do something else?
- Games can differ in a number of ways:
- Zero-sum or non-zero sum:
- Zero-sum game: complete conflict – one player’s gain is
another player’s loss.
e.g. advertising that affects market share but not market
size; ‘Matching pennies’ game (later handout)
- Non-zero sum – different combinations of strategies alter the
size of the combined payoffs.
e.g. advertising that changes market size as well as
shares; most examples below are non-zero sum.
- Economic interactions e.g. market exchange: zero-sum or not?
- Simultaneous or Sequential games:
- Are player moves made at the same time or sequentially?
- One-time (Static) or Repeated (Dynamic) games?
- Will the game be repeated or not?
- If repeated is there a known final period?
- Complete or Incomplete Information
e.g. how much information about other players payoffs do
players have? how much information about own payoffs?
- Is communication between the players possible?
- Specific, well-known games are defined by relative payoff sizes for
different combinations of “moves” (special strategic cases)
e.g. Prisoner’s Dilemma, Chicken, Battle of the Sexes etc. are defined by comparative size of the payoffs (see below)
Oligopoly and Game Theory:
- Oligopoly as a game:
- Players: businesses in the oligopoly
- Strategies: could be output (like Cournot); price (like Bertrand); other business decisions (expand or not? advertise or not? location?)
- Payoffs: profits of the firms in the oligopoly.
- Decision rules: play best responses.
- A common theme in oligopoly models is the tension between collusion and
competition.
- Collusion: oligopolists cooperate, limit competition to raise profits.
Extreme cooperation: cartel
- oligopolists act collectively like a monopolist.
- Some oligopolistic markets exhibit collusive behavior, others
competitive behavior (e.g. Boeing vs. Airbus), others exhibit both at different times.
- The decision to compete or collude may be made operational
through a variety of strategy variables e.g.,
- Pricing decision (as in the example below): high or low price?
- Output decision.
- Investment in capacity to produce output.
- Advertising decisions
- Location decision: should you open a branch in the
competitor’s market?
A One-Time Game theoretic model of oligopoly:
- Players: MegaBiz (Firm 1), Giantcorp (Firm 2). (duopoly)
- Strategies:
Each firm can either charge a high price (collude) or low price (compete):
(1) Collude (cooperate); (2) Compete (defect from
e.g. cartel. cooperation).
- Payoffs: profits received by each player
- Megabiz (Firm 1) payoff: p1 Giantcorp (Firm 2) payoff: p2
- Typically these are just given in a payoff matrix or decision tree but
here lets derive them.
- Collude? - maximize joint profits, i.e., act as if they were a monopoly.
- Say that the market demand curve is:
P = 10 – 0.1Q (text example different numbers)
then:
MR = 10 – .2Q
if the firms act like a monopolist they set output so that:
MR = MC
- assume constant MC = 2
then joint output should be:
10 – .2Q = 2 ® Q = 40
and price:
P = 10 – .1Q = 6
Joint profit is: P Q – mQ = 6x40 – 2x40 = 160
(assume a fixed cost of = 0 and constant MC)
- If the duopolists split the market the payoff is 80 for
each firm (p1 =80 , p2 =80 ).
- Compete (defect):
- Say that a firm breaking the collusion agreement undercuts the
agreed price of 6, e.g., charges 5.
- Payoffs from defecting?
- Depends on the action of the other firm.
- Say MegaBiz sets P=6 and Giantcorp sets P=5.
- no sales for MegaBiz so p1 = 0.
- Giantcorp gets entire market:
5 = 10 – .1Q
Q = 50 p2 = 5x50 – 2x50 =150
- Say MegaBiz sets P=5 and Giantcorp sets P=6.
- no sales for Giantcorp so p2 = 0.
- MegaBiz gets entire market: p1 = 150
- Say both MegaBiz and Giantcorp set P=5 (compete!)
5 = 10 – .1Q Q = 50
split the market and profit: output 25 each
p1 = 75
p2 = 75
Summarizing strategies and payoffs: “Payoff matrix”
MegaBiz (1)
Collude Defect
(P=6) (P=5)
Collude p1 = 80 p1 =150
Giantcorp (2) (P=6) p2 = 80 p2 = 0
Defect p1 = 0 p1 = 75
(P=5) p2 = 150 p2 = 75
(Note: - the pattern of payoffs makes this a Prisoner’s Dilemma game
- is it a zero-sum game? )
- “Strategic” or “Normal” form: game in terms of a payoff matrix.
- Alternative? “Game tree” or ‘Extensive form’ – often used in Sequential
Games.
(see the oligopoly game in extensive form next page)
- Decision rule: assume each chooses their highest payoff response to their
opponent’s choice.
- Solving: first find ‘best responses’ to each possible choice.
- Consider MegaBiz’s decision:
- if Giantcorp Colludes:
MegaBiz gets 80 if it also colludes.
MegaBiz gets 150 if it defects.
So Megabiz should Defect!
- if Giantcorp Defects:
MegaBiz gets 0 if it colludes.
MegaBiz gets 75 if it defects.
So Megabiz should Defect!
- Note: above gives “Best response” to opponents choice (like a
reaction function)
- Defect is the best strategy regardless of what Giantcorp
does: this is a dominant strategy.
(Collude is a “dominated strategy” – if there are 3+ strategies can have
“dominated strategies without there being a dominant strategy)
- Consider Giantcorp:
- Giantcorp’s problem is identical to that of MegaBiz.
- Defect is also a ‘dominant strategy’ for Giantcorp.
- Likely equilibrium solution to the game:
- both firms play their dominant strategy (Defect).
- outcome: p1 = 75 p2 = 75
- When players have dominant strategies a plausible equilibrium involves
each firm playing its dominant strategy.
- Collusion vs. competition: in oligopoly
- the two firms would both have higher profit if they both collude
rather than both defect.
- An agreement to collude would benefit both firms.
- problem: both parties have an incentive to cheat on the
agreement!
- unless there is some way to “enforce” the agreement collusion
is likely to break down.
- Creating cooperation (collusion):
- Legal contract could solve the problem.
- Both parties agree to collude.
- Penalties to defection are specified and enforced by some
third-party (courts?).
- These penalties change the payoffs to the defect strategy.
- If the penalties to defection are large enough collusion
becomes the better strategy.
- But: in many countries such agreements are illegal (e.g.
The Competition Act in Canada).
- “under-the-table”, informal agreements have the
problem of non-enforceability.
- More generally “punishments” and “rewards” could create
cooperation.
e.g. via behavior in other situations where the same players
interact.
- Price matching policies: “will match or beat any competitor’s price”
- Will this move the outcome to “collusion”?
- Repeated games: if the situation occurs repeatedly will collusion be
more likely? (see discussion below)
- future opportunities to punish defection and reward collusion.
- The Oligopoly game above is one example of a Prisoner’s Dilemma game.
- There are many other situations of strategic interaction that are also
Prisoner’s Dilemmas.
- Generally situations where cooperation gives a better outcome than
not cooperating. Problems of collective action.
Examples:
- the free rider problem and provision of a collective good.
- conservation and a common property resource (Tragedy of the
Commons)
- international climate change policy: is it a Prisoner’s
Dilemma?
Nash Equilibrium:
- Nash equilibrium is a key equilibrium concept in game theory.
- Nash equilibrium:
Each player chooses their best strategy given the strategies chosen
by the other players.
- with a Nash equilibrium no player has an incentive to change their
strategy.
- so a Nash equilibrium is likely to be stable.
- The equilibrium in the duopoly game above is a Nash equilibrium.
MegaBiz’s best choice is defect given that Giantcorp chooses defect.
Giantcorp’s best choice is defect given that MegaBiz chooses defect.
- any game where all players are playing dominant strategies is a Nash
equilibrium !
- but Nash equilibrium need not have players playing dominant
strategies.
Example: Game with a Nash Equilibrium but one firm has no
dominant strategy (see Table 13-6, handout)
Example: Dixit and Nalebuff’s oligopoly pricing example
- Handout: several possible prices (strategies).
- Solving for Nash equilibria:
- For each player: identify the “best” choice for each of the
opponent’s choices.
- Are there any “overlapping” best choices: these are Nash
equilibria.
Example: A game with no dominant strategies but multiple Nash equilibria
Player 1
Plan A Plan B
Plan A p1 = 40 p1 =20
Player 2 p2 = 21 p2 =15
Plan B p1 = 20 p1 = 30
p2 = 10 p2 = 20
- no dominant strategies.
- combinations: Plan A, Plan A
Plan B, Plan B
- Are both Nash equilibria.
- Is there any reason to expect one to occur rather than the
other?
- here Plan A, Plan A seems likely: both prefer it to B,B.
- But what if Player 1 only gets 29 (not 40) from A,A?
- Many “Coordination games” are similar to the game above:
- Players do better if they coordinate.
- But there are multiple Nash equilibria.
- “Battle of the Sexes”, “Chicken”, Dixit and Nalebuff’s
Hunters (handout), Driving game (which side of road).
- Major question is whether there is any reason to expect one
over the other?
- is there a reason outside the model to expect a specific
equilibrium? e.g. are there “focal points”
- Binmore: trial and error could mean that any Nash
Equilibrium could prevail (if more than one).
(NOTE: Movie “Beautiful Mind” about John Nash --- Nash’s suggested
solution in the bar scene is not a Nash equilibrium!)
- The appeal of Nash equilibrium?
(1) plausible that rational players will reason their way to this type of
outcome;
(2) evolutionary justification: will players facing these situations
evolve toward a Nash equilibrium and then stay there?
e.g. do only the most successful players survive in the long run?
Will experimentation eventually reach then stay at this
outcome? (even in complex games)
(3) self-enforcing: no incentive to change behavior once there.
Repeated Games:
- All the games above are “one-time” or one period games.
- What happens if the players know that the game will be repeated several
times?
- Go back to the Megabiz-Giantcorp oligopoly game above:
- solution was for both players to defect.
- defecting gives a lower payoff than colluding.
- if the game is repeated there seems to be more of an
incentive to find some way to collude.
- otherwise receive the lower payoff repeatedly if they
always defect.
- new strategic possibilities: use threats or promises about
future interactions to influence opponent today.
e.g. threaten to punish ‘bad behavior’ via future actions.
- perhaps oligopoly is more sensibly modeled as a repeated
game? Oligopolists co-exist for many periods.
Strategies to induce collusion in repeated games:
- Megabiz must make it clear to Giantcorp that there are negative
consequences to defection.
- assume again that a legal contract is not possible
- then MegaBiz must “punish” Giantcorp for defection.
- in the simple two-strategy game MegaBiz can punish by
defecting as well.
- There are many possible strategies. A much-discussed possibility?
Tit-for-tat
- MegaBiz plays collude (C) in the first round. Then in subsequent
rounds MegaBiz does whatever Giantcorp did in the previous round.
e.g., Round 1 Megabiz plays C
- if Giantcorp plays C then MegaBiz plays C in round 2.
- if Giantcorp plays Defect (D) then MegaBiz plays D in
round 2. (punishment)
- If both are playing “tit-for-tat”: collude forever
- If Giantcorp only plays ‘D’ it wins in period 1 but never wins again:
this could limit the attractiveness of playing D.
- Tit-for-tat: very successful in tournaments and computer simulations of
repeated games against a variety of opposing strategies.
- Advantages?
Clear – simple for other players to infer that the opponent
is playing it.
Nice - starts playing C.
Tough – punishes D immediately.
Forgiving – reverts to C if the other player does.
- Repeated games and reputation.
- If a player is going to repeatedly play the same game or will play
similar games with other players it may “invest” in reputation.
i.e., accept lower payoffs now in order to establish a reputation that
will permit larger payoffs in later games.
- Unravelling Problem in Finite Repeated Games:
- Unravelling occurs in repeated games with known end period: