• An industry's "Performance"
  • Price (high, low)
  • Output (high, low)
  • Profit
  • Productive efficiency
  • Allocative efficiency (CS, DWL)
  • Quality, safety, etc.

• ... which is determined by its firms' "Conduct"
  • Pricing behavior
  • Output choices
  • Legal tactics
  • Entry and exit
  • Collusion
  • Competition
  • Innovation

• ... which is determined by the industry's "Structure"
  • Number of buyers
  • Number of sellers
  • Product differentiation
  • Vertical integration
  • Barriers to entry

• ... which is determined by the product's "Technology"
  • Consumer demand (price and income elasticity, substitutes, location, etc)
  • Good's features: inputs, technology, durability, rivalry, excludability
  • Production features: economies of scale, economies of scope, network externalities, two-sided platforms

• Government policy can affect nearly every stage
  • Entry regulation
  • Price regulation
  • Antitrust regulation
  • Taxes and subsidies

• As explore other aspects of firm behavior, we need more tools from game theory beyond a one-shot prisoners' dilemma

• We've seen examples of continuous strategies in Cournot, Bertrand, & Stackelberg competition

  • Firms can choose a wide range of prices, outputs
  • One-shot prisoners' dilemma we had two discrete choices

• Seen simultaneous (prisoners' dilemma) vs. sequential game (Stackelberg)

• Assumed common knowledge about the game structure. Everyone knows all players, all their potential strategies, and all of their payoffs

• Perfect information (players can observe moves of others, as in sequential game) vs. Imperfect information players cannot observe moves of others (Prisoners' Dilemma)

• We use "solution concepts" to allow us to predict an equilibrium of a game

• Nash Equilibrium is the primarly solution concept

  • Note it has many variants depending on if games are sequential vs. simultaneous, perfect vs. imperfect information, etc.

• Recall, Nash Equilibrium: no players want to change their strategy given what everyone else is playing
  • All players are playing a best response to each other

• Important about N.E.:
  • N.E. ≠ the "best" pr optimal outcome
  • Recall the Prisoners' Dilemma!
  • Game may have multiple N.E.
  • Game may have no N.E. (in "pure" strategies)

• We consider an Entry Game, a sequential game played between a potential Entrant and an Incumbent

• A sequence of play: Entrant moves first, Incumbent moves second

• Note: the magnitude of the payoffs don't really matter, only their relative sizes

  • Hence, my simple numbers

• This game is depicted in "Extensive form" or a game tree

• Each player faces at least one "decision node" (solid, colored by player)

  • Entrant chooses between Enter or Stay Out at node E.1
  • Incumbent chooses between Accommodate or Fight at I.1
  • Game ends at any "terminal node" (hollow node), and each player earns payoffs (Entrant, Incumbent)

• We need to talk more about strategies

• Pure strategy: a player's complete plan of action for every possible contingency

  • i.e. what a player will choose at every possible decision node
  • think like an algorithm:

if we reach node 1, I will play X; if we reach node 2, I will play Y; if ...

• "Mixed strategy": play a strategy with some probability

• Entrant has 2 pure strategies:

  1. Stay Out at E.1
  2. Enter at E.1

• Incumbent has 2 pure strategies:

  1. Accommodate at I.1
  2. Fight at I.1

• Note Incumbent's strategy only comes into play if Entrant plays Enter and the game reaches node I.1

• Backwards induction: to determine the outcome of the game, start with the last-mover (i.e. decision nodes just before terminal nodes)

• What is that mover's best choice to maximize their payoff?

• i.e. we start at I.1 where Incumbent can:

  • Accommodate to earn 1
  • Fight to earn 0

• Incumbent will Accommodate if game reaches I.1

• Given this, what will Entrant do at E.1?

  • Stay Out to earn 1
  • Enter, knowing Incumbent will Accommodate, and so will earn 2

• Entrant will Enter

• Continue until we've reached the initial node (beginning)

• We have the outcome:

• Some textbooks call this a "rollback equilibrium"

• Any game in extensive form can also be depicted in "normal" or "strategic" form (a payoff matrix)

• Note, if Entrant plays Stay Out, doesn't matter what Incumbent plays, payoffs are the same

• Solve this for Nash Equilibrium...

• Solve this for Nash Equilibrium...

• There's no dominant strategies!

• Here's two methods that will help:

1 - Cell-by-Cell Inspection: look in each cell, does either player want to deviate?

• If No: a Nash Equilibrium
• If Yes: not a Nash Equilibrium

2 - Best-Response Analysis: take the perspective of each player. If the other player plays a particular strategy, what is your strategy(s) that gets you the highest payoff?

• Ties are allowed
• Any cell where both players are playing a best response is a Nash Equilibrium

• Two Nash Equilibria:

1. (Enter, Accommodate)
2. (Stay Out, Fight)

• But remember, we ignored the sequential nature of this game in normal form

  • Which Nash equilibrium is "sequentially rational?"

• New solution concept: "subgame perfect Nash equilibrium" (SPNE)

• Subgame: any portion of a full game beginning at one node and continuing until all terminal nodes

  • i.e. any decision node starts a subgame containing all the "branches" of that decision node

• Every full game is itself a subgame

• How many subgames does this game have?

• Suppose I were to announce that if you were late once to class, I gave you an F

• If you believe my threat, you would arrive on time, and I never have to carry out my threat

• Sounds like a Nash equilibrium...but not subgame perfect!

• If you call my bluff and come late, I don't actually want to carry out my policy!

• "Talk is cheap"

• With perfect information, promises or threats will not change equilibrium if they are not credible

  • Must be "sequentially rational" or SPNE strategy

• Strategy must be incentive-compatible, if game reaches the relevant node, it must be in your interest to carry out your promise or threat!

• Threats and promises can be credible with commitment

• A commitment changes the game in a way that forces you to carry out your promise or threat

  • tying your own hands makes you stronger!

• Suppose that before Entrant can decide to Enter or Stay Out, Incumbent can choose to Invest in excess capacity

• i.e. builds up a "war chest" of superfluous infrastructure that costs -1

• But with it, in response to Entrant playing Enter, Incumbent is in a better position to Fight a price war that kicks Entrant out of the market

• Game changes, Incumbent goes first at (new) I.1, deciding whether to Invest or Don't

  • Game is the same as before from E.2 onwards

• This is a more complicated game, let's apply what we've learned...

• What are the subgames?

• What are the subgames?

1. Subgame initiated by node I.1 (game itself)
2. Subgame initiated by node E.1
3. Subgame initiated by node E.2
4. Subgame initiated by node I.2
5. Subgame initiated by node I.3

• What are the strategies available to each player?

• What are the strategies available to each player?

• Entrant, choosing at nodes (E.1, E.2)

  1. (Stay Out, Stay Out)
  2. (Stay Out, Enter)
  3. (Enter, Stay Out)
  4. (Enter, Enter)

• What are the strategies available to each player?

• Incumbent, choosing at nodes (I.1, I.2, I.3)

  1. (Invest, Accommodate, Accommodate)
  2. (Invest, Accommodate, Fight)
  3. (Invest, Fight, Accommodate)
  4. (Invest, Fight, Fight)
  5. (Don't, Accommodate, Accommodate)
  6. (Don't, Accommodate, Fight)
  7. (Don't, Fight, Accommodate)
  8. (Don't, Fight, Fight)

• We can use backwards induction to find the outcome of the game

• Start with best response of Incumbent at I.2 and I.3...then best response of Entrant at E.1 and E.2...then Incumbent at I.1

• Define the outcome in terms of their strategies:

{ (Invest, Fight, Accommodate), (Stay Out, Enter) }

• The equilibrium path of play is Invest, then Stay Out

• Note via backward induction, we've shown the optimal response by each player at each decision node

{ (Invest, Fight, Accommodate), (Stay Out, Enter) }

• SPNE: this set of strategies induces a Nash equilibrium in every subgame

  • Each player optimally responding to every contigency

• With commitment, it is credible for Incumbent to threaten to Fight if Entrant decides to Enter!

{ (Invest, Fight, Accommodate), (Stay Out, Enter) }

• Note we could look at the set of strategies in normal form

• Using best response, there are many Nash Equilibria (boldened)!

  • But only the one we mentioned is subgame perfect!