Game Theory and Models of Oligopoly

- Main Source: text Ch. 12 (new edition), Ch. 13 (older editions)

- 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? What about collusion vs. competition?

- 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.

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- 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) assuming others do the same.

- what to do if opponents’ choices are uncertain or

possibly irrational? Less clear.

e.g. ‘maximin’ seek to maximize your minimum possible payoff (a risk-averse rule)

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- Games can differ in a number of ways:

- 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?

- Complete or Incomplete Informatione.g. how much information about other players payoffs do

players have? how much information about own payoffs?

- Evolutionary games: population of players evolves (those playing successful strategies become more common)

- Specific, well-known games are defined by relative payoff sizes for different combinations of “moves” which give rise to special strategic cases.

e.g. Prisoner’s Dilemma, Chicken, Battle of the Sexes etc. are defined by comparative size of the payoffs (see

below)

- Zero-sum or non-zero sum: a distinction based on total payoff size

- Zero-sum game: one player’s gain is another player’s loss.

e.g. advertising that affects market share but not market size;

- Non-zero sum – different combinations of strategies alter the size of the combined payoffs.

e.g. advertising that changes market size and market shares.

- Economic interactions e.g. market exchange: zero-sum or not?

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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

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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: 1 Giantcorp (Firm 2) payoff: 2

- 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 and 0 fixed cost then joint output is:

10 – .2Q = 2 Q = 40 and price:

P = 10 – .1Q = 6

Joint profit is: P Q – mQ = 6x40 – 2x40 = 160

- If the duopolists split the market the payoff is 80 for

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each firm (1 =80 , 2 =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 1 = 0.

- Giantcorp gets entire market:

5 = 10 – .1QQ = 50 2 = 5x50 – 2x50 =150

- Say MegaBiz sets P=5 and Giantcorp sets P=6.

- no sales for Giantcorp so 2 = 0.

- MegaBiz gets entire market: 1 = 150

- Say both MegaBiz and Giantcorp set P=5 (compete!)

5 = 10 – .1Q Q = 50

split the market and profit: output 25 each

1 = 75 2 = 75

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Summarizing strategies and payoffs: “Payoff matrix”

MegaBiz (Gets 1)

Collude Defect (P=6) (P=5)

Collude 1 = 80 1 =150Giantcorp (P=6) 2 = 80 2 = 0 (gets 2)

Defect 1 = 0 1 = 75 (P=5) 2 = 150 2 = 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)

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- 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!

- This exercise gives “Best responses” 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)

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- 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 ‘Defect’

- outcome: 1 = 75 2 = 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.

- See text: “Commitment problem” (p. 414)

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- 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 enforceability problems.

- More generally “punishments” and “rewards” could create cooperation.

e.g. via behavior in other situations where the same players interact.

(Comment: Competition Bureau uses penalties to do the opposite: to make collusion unattractive)

- 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.

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- Behavioral assumptions:

- is the dilemma only a problem for ‘Homo Economicus’

- will more trusting people or societies solve such problems? e.g. if they feel that defection is immoral.

- The Oligopoly game above is a Prisoner’s Dilemma (PD) game.

- There are many other situations of strategic interaction that are also Prisoner’s Dilemmas, text Table 12.3 (Appendix), 12.6, concept check 12.1)

- Generally it’s a PD game when cooperation gives a better outcome than both not cooperating but incentives to ‘defect’ from cooperation exist.

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?

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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 12-4 (13-6 in old

edition, Appendix)

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Example: Dixit and Nalebuff’s oligopoly pricing example (Appendix)

- 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 1Plan A Plan B

Plan A 1 = 40 1 =20Player 2 2 = 21 2 =15

Plan B 1 = 20 1 = 30 2 = 10 2 = 20

- no dominant strategies.

- combinations: Plan A, Plan APlan 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:

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- Players do better if they coordinate.

- But there are multiple Nash equilibria.

- “Battle of the Sexes”, “Chicken”, Dixit and Nalebuff’s Hunters (Appendix), 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:

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- 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.

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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.

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- 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:

- say that both players know that the repeated Prisoner’s dilemma game will end next period.

- now it is a one-period game with equilibrium: D,D.

- given that players will play D,D in the last period what should they do in the second last period?

- play D,D !

- working backwards parties will always defect in a finite repeated Prisoner’s dilemma game.

- only when the number of rounds is unknown will tit-for-tat or other strategies be able to induce collusion.

(this is a common issue of repeated games)

- Cooperation in repeated P.D. games: when is it likely?

- no known end period (so no unravelling)

- players must care sufficiently about future payoffs (discount rate can’t be too high)

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Sequential Games:

- Decisions were treated as simultaneous above.

- Decisions may be sequential instead: One player takes an action, the other responds.

- Sometimes it doesn’t matter if the game is sequential not simultaneous.

- say that in the one-period game MegaBiz moves first and Giantcorp responds.

- MegaBiz chooses its dominant strategy anyway.

- Giantcorp chooses its dominant strategy in response.

- Same result.

- Table 12-4 (13-6): a game where moving first changes the outcome: if Firm 1 moves first it chooses Don’t advertise Firm 2 then chooses Don’t advertise.

(vs. the Nash equilibrium Advertise-Advertise if simultaneous)

- “Game trees” or “decision trees”: often useful in illustrating sequential games.

- first branches: possible moves of the first mover.

- at the end of each branch new branches for each possible response of the other player.

- at the end of each set of decision-response branches the payoffs for each player.

- Solving sequential games: “Look forward, reason back”.

(Survivor Flag game example)

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Oligopoly Application: Entry Deterrence as a Sequential Games

- Imagine one firm is already in a market but there is potential for entry by a competitor.

- Can the firm already in the market take actions to deter entry?

- Text: Tallest building example Figures 12-12 or Amusement park example in old edition (Figure 13-7).

- building a platform or purchasing land now alters payoffs in such a way to make entry unattractive.

- Can adding capacity signal an intention to flood the market with output should entry occur?

- Can creating output stockpiles (inventories) do the same thing? Debeers and diamond stockpiles (see Varian).

- Limit pricing model:- Say firms behave as in the Stackelberg model (Leader:

incumbent), follower (potential entrant).

- entrant assumes the existing firm will continue acting as it does.

- With fixed costs to entering the “Leader” can produce high output (set low price) now and deter entry.

- Note: even the potential for entry drives the outcome toward the competitive outcome i.e., more output and lower price than monopoly.

(“Contestable Markets”: similar idea discussed in text)

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Solving for the Outcome in Game theoretic models: a summary

- First determine “best responses” of each player to the other players’ actions.

- Then (Dixit and Nalebuff):

- Is there a dominant strategy? (they will be used)

- Are there dominated strategies? (eliminate them)

- Check if there are any Nash equilibria? (good candidate solutions)

- Sequential games: look forward then reason backwards.

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Game theory and Rationality:

- Are decision-makers as rational as game theory suggests?

e.g. Dixit and Nalebuff: “Half-way”, “Survivor’s Flag game”

- Experimental evidence: - some limitations suggested especially as games become

more complex.

- but do “experts” (successful firms, professionals) do better?

- experiments usually show learning occurs if games are redone.

- If facing an irrational opponent what do you do? ‘Maximin’ a possibility?

- Complex games and large numbers of possible payoffs and strategic interactions: can players work out solutions?

e.g. chess

- Decision making across time and with uncertainty: games can be quite complex issues.

- Equilibria in games: two interpretations

- Rational interpretation: players work out the games (as above), identifying best strategies and likely outcomes.

- Evolutionary interpretation: players choose strategies, results are observed. Less fit (low payoff) strategies are gradually displaced.

i.e. evolve toward rational outcomes.

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- Are decision-makers as self-interested as economic models usually

assume?

- are they more altruistic? Are they concerned about fairness?

- some examples from Dixit and Nalebuff: contribution game; “satisfaction from punishing cheaters”; Ultimatum game and fair divisions.

- Flexibility.

- Often in game theory the solution is ambiguous.

- Does this fit well with actual markets?- many types of oligopoly outcomes observed.

(Note: all the examples above use “Pure strategies” – alternative is “Mixed strategies” : mixed strategy responses are purposely randomized

e.g. choose strategy A with probability 0.5 and strategy B with probability 0.5 (say on basis of a coin flip)

Penalty kicks in soccer: mixed strategies by shooter and goalkeeper are the Nash equilibrium. )

- Take the Game theory course next year for all games all the time.

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APPENDIX: SOME EXAMPLE GAMES

(New edition: Table 12-3)

(New edition Table 12-4)

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(Schelling The Strategy of

Conflict)

(Binmore Game Theory: a Very Short Introduction)

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(Binmore Game Theory: a Very Short

Introduction

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A Pricing game from Dixit and Nalebuff Games of Strategy:

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Source: Dixit and Nalebuff

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