Managerial Economics MBACatólica
Fernando Branco
2006-2007 Fall Quarter Session 7
Strategic behavior ♦Strategic behavior is very relevant in situations other than oligopoly markets: – Bargaining among employers and unions; entry and exit in markets; R&D and product innovation; ... ♦Game Theory provides the tools for the analysis of strategic issues. ♦Example: Setting quotas at OPEC.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Setting quotas at OPEC In 1990s OPEC found it difficult to maintain high prices. Some countries, such as Venezuela often produced more than agreed. In the mid-nineties Venezuela asked for a restriction in quotas, but Saudi Arabia (who usually supports low productions) refused it. Analists predicted a stronger cartel for the following years. Why?
Adapted from Baye, 1997 (pp. 361).
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
1 Game Theory ♦Game theory provides techniques to analyze and describe the behavior of agents in strategic interaction situations. ♦Used in many areas, and it has changed the approach to many problems in Economics and Management .
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
A typology of games
♦What is a game?
♦Static games vs dynamic games.
♦Games of complete information vs games of incomplete information.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
What is a game?
♦A game is a situation of strategic interaction among several agents: – The decision of a player affects the outcomes of the other players. ♦Examples of games: tic-tac-toe, chess, bridge, Cournot duopoly, political parties and electors.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
2 Static games vs dynamic games
♦In some simple games, players play once and simultaneously: these are the static games. – Scissors-paper-stone ; Bertrand duopoly. ♦In some other games, players react to each others decisions: these are dynamic games. – Draughts; Stackelberg duopoly.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Games of complete information vs games of incomplete information
♦ In some games all players know the same about the game: they are games of complete information. – Chess. ♦ In some other games, some players have different information about the game: They are games of incomplete information. – Bridge.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Elements of a game
♦ Players
♦ Strategies
♦ Outcomes
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
3 Elements of a game
♦ Players ♦ Agents that take the decisions. ♦ Strategies – managers, customers, workers, government, voters, etc. ♦ Outcomes
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Elements of a game
♦ Players ♦ Actions that can be taken by each player in any possible ♦ Strategies situation of the game: – Rules of the game. ♦ Outcomes
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Elements of a game
♦ Players ♦ Result achieved by each player at the end of the game: ♦ Strategies – Win/loss; points; profits; votes.
♦ Outcomes
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
4 How to describe static games?
♦Static games with two players and a small number os possible actions can be described through a table (the payoff matrix). ♦Static games with more than two players or a larger number of possible choices are described in a different way.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Payoff matrix
Advertising game
Firm B Yes No Firm Yes 10 , 5 15 , 0 A No 6 , 8 10, 2
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Playing a game ♦Dominated and dominant strategies. ♦Best reply. ♦Nash equilibrium.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
5 Playing a game ♦Dominated and ♦Dominated strategy : Yields dominant payoffs that are always worse strategies. than those of some other ♦Best reply. strategy. ♦Nash ♦Dominant strategy : Yields equilibrium. payoffs that are better than those of any other strategy.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Playing a game ♦Dominated and dominant ♦Players should not play strategies. dominated strategies and ♦Best reply. should choose dominant strategies. ♦ Nash ♦Examples. equilibrium.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Dominated and dominant strategies (I)
Advertising game Firm B Yes No Firm Yes 10 , 5 15 , 0 A No 6 , 8 10 , 2
‘Yes’ is a dominant strategy: Both firms should advertise.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
6 Dominated and dominant strategies (II)
♦If there is a dominant strategy the others are dominated strategies. ♦But a strategy may be dominated and without existing a dominant strategy. Firm B High Low No Firm Yes 10 , 5 12 , 4 15 , 0 A No 6 , 8 7 , 9 10, 2
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Dominated and dominant strategies (III)
♦Some games do not have dominated nor dominant strategies.
Firm B Market 1 Market 2 Firm Market 1 –1 , –1 2 , 3 A Market 2 3 , 2 –2 , –2
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Playing a game ♦Dominated and dominant ♦What is the best action of a strategies. player, given the actions of ♦Best reply. the other players? ♦Nash ♦Examples. equilibrium.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
7 Best replies
Advertising game
Firm B Yes No Firm Yes 10 , 5 9 , 6 A No 6 , 8 10 , 2
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Playing a game ♦Dominated and dominant strategies. ♦All players are choosing best replies to the other players. ♦Best reply. ♦ ♦Nash Examples. equilibrium.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Nash equilibrium (I) Choice of prices Firm B Low High Firm Low 0 , 0 3 , –1 A High –1 , 3 2 , 2 ♦Nash equilibrium: (Low, Low). ♦Note that (High, High) would yield higher payoffs. Why is it not an equilibrium?
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
8 Nash equilibrium (II) Entry in new markets
Firm B Market 1 Market 2 Firm Market 1 –1 , –1 2 , 3 A Market 2 3 , 2 – 2 , – 2
♦There may be more than Nash equilibria.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Nash Equilibrium (III) Scissors-Paper-Stone Player B Scissors Paper Stone Scissors 0,0 1,−1 −1,1 Player A Paper −1,1 0,0 1,−1 Stone 1,−1 −1,1 0,0
♦ There may be no (pure-strategy) Nash equilibrium.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Continuous choices ♦Some games require a choice from of a continuous variable (e.g., price). ♦One can not use a payoff matrix to decribe the game. – Describe the strategies; – Provide a payoff function for each player. ♦Example: Parties and the political proposals
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
9 Parties and the political proposals
♦The political agenda can be described as selecting a index between 0 (left) and 1 (right); ♦Electors are uniformly distributed in the political spectrum. ♦A party payoff is his number of votes.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Parties and the political proposals (II)
♦What is the equilibrium, if there are two parties?
♦And what if there are four parties?
♦And waht if there are three parties?
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Nash equilibria in games with continuous actions ♦In games with continuous actions, we should proceed as we did in the Cournot oligopoly: – Identify best replies; – Solve them togheter.
♦Or, as we did in the Bertrand oligopoly: – Think about the dynamics of best replies.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
10 Dynamic games ♦We will look at two kinds of dynamic games. – Sequential games; – Repeated games. ♦It will also be important to distinguish among games with: – A predetermined final period; – An unknown final period.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Sequential games
♦The players decide one at a time (often alternating): – Tic-tac-toe, chess, Stackelberg duopoly;
♦Payoffs are determined at the end.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Repeated games
♦A static game is played in every period. ♦All players observe the past choices. ♦Payoffs are (discounted) sums of per period payoffs.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
11 Equilibria in dynamic games
♦Nash equilibrium: – Each player is doing the best he cans, given the choices of the other players.
♦Subgame perfect equilibrium: – More reasonable Nash equilibrium.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Subgame perfect equilibrium
♦Notion of subgame. ♦Subgame perfect equilibrium: – The idea of Nash equilibrium is applied to all subgames; – The prescribed play is optimal for every possible contingency in the game.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Sequential games
♦How to describe a sequential game (extensive form)?
♦Equilibria: – Backwards induction (yields subgame perfection); – Other Nash equilibria.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
12 How to describe a sequential game?
♦Tree-diagrams: – Nodes: decision moments; – Branchs: possible actions. ♦Example of a sequential game
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Tree diagrams
♦ These are similar diagrams used in probability trees or decision trees. ♦ It is just a sequence of nodes and branchs: each branch starts in one node and ends in another node.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Example of sequential game (10,15) ♦ up Player A starts by B choosing between p d “Up” and “Down”. U ow n ♦ Player B then A (5,5) chooses between (0,0) Down p “up” and “down” B u ♦ Payoffs are at the d end. ow n (6,20)
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
13 Equilibria in sequential games
♦ Proceeding backwards from the end of the tree. ♦ Identify the action that will lead to the highest terminal payoff. ♦ This will yield a subgame perfect equilibrium. ♦ There may be other Nash equilibria: – The role of on-credible threats. ♦ Revisiting the Stakelberg duopoly game
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Backwards induction in a sequential game
p (10,15) ♦ B chooses “up” B u after “Up” and p d “down” after U ow A n “Down”. (5,5) ♦ A chooses “Up” (0,0) Down p ♦ This is the unique B u subgame perfect d equilibrium. ow n (6,20)
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Non-credible threats and another Nash equilibrium p (10,15) ♦ B chooses “down” B u regardless of A p d previous choice. U ow A n (5,5) ♦ A chooses “Down” Down (0,0) p B u ♦ This is an somehow d unpleasant Nash ow equilibrium. n (6,20)
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
14 Revisiting the Stackelberg duopoly
♦The equilibrium described for the Satckleberg duopoly is a subgame perfect equilibrium. ♦However, the outcome of a Cournot duopoly might be achieved in a Stackelberg duopoly. How?
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Sequential games with unknown final period
♦If the game is potentially infinite, backwards induction is not straightforward to apply: – A final period can not be identified.
♦Example: A bargaining game
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
A potentially infinite bargaining game ♦Two players need to decide how to share a pie; ♦Player A starts my proposing a division; ♦Player B may accept (if so the game ends) or reject (if so, they will meet again in the following period where B makes an proposal). ♦The game is potentially infinite. ♦What is the equilibrium?
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
15 Properties of the equilibrium
♦There is a first move advantage. ♦Players benefit from being more patient.
♦The first proposal is accepted!
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Repeated games with known end-period
♦ Describing such a game. ♦ The repeated Prisoners’ Dilemma Game . ♦ Basic structure of the equilibria: – The effect of the final period. ♦ Subgame perfect equilibria: – Repeating Nash equilibria of the period game; – Other equilibria.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
The repeated Prisoners dilemma game
♦The Prisioners’ Dilemma Game Firm B Yes No Firm Yes 10 , 5 15 , 0 A No 6 , 8 10, 2 ♦What will happen if the game is played twice? And a thousand times?
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
16 A unique equilibrium
♦Play always as in the static game!
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Efect of the final period
♦In the last period, players should chose as in the static game; ♦Behaving like in every period that provides common equilibria.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Other equilibria (I)
♦Consider the following base game: Player B LMR Player T 2 , 4 2 , 2 1 , 3 A B 1 , 1 5 , 1 4 , 2
♦In the last period play will be either (T,L) or (B,R). ♦Can (B,M) be observed in the first period?
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
17 Other equilibria (II) ♦ For (B,M) to be observed in period 1, player B needs t be prevented from chosing R (improving 1). ♦ How can this be done? – Making the choice of R by player B in period 1, bring a greater loss in period 2. ♦ This can be achieved with the following strategies: – Player A: choose B in period 1; in period 2, choose T if (B,M) has been played in period 1, but B otherwise; – Player B: choose M in period 1; in período 2, choose L if (B,M) has been played in period 1, but R otherwise.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Infinitely repeated games
♦There is no final period. ♦Many subgame perfect equilibria: – Based on actions that do not support a Nash equilibrium of the static game (trigger strategies). ♦The relevance of infinite games. ♦Application: Collusion in oligopolies.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Infinite repetition of the Prisioners’ Dilemma Game ♦Base game: Prisoners’ dillema Player B LR Player T 5 , 5 15 , 0 A B 0, 15 12, 12 ♦If the game is played infinitely (or with unknown final period) will it be possible to sustain (B,R) in each period?
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
18 Equilibrium with (B,R) in each period ♦Infinite play on (B,R) yields to each player: ∞ − 12 12 δ t 1 = ∑ −δ t=1 1 ♦Alternatively, a player obtains: ∞ 5δ 15 + 5δ t =15 + ∑ −δ t=1 1 ♦Equilibrium in (B,R) if: 12 5δ 3 ≥15 + ⇔ δ ≥ 1−δ 1−δ 10
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
The relevance of infinite games
♦In a sense every game is finite. ♦However, most business interactions are happen repeadly, but with unknown final period: – This makes the game similar to an infinite game.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Colusion in oligopolies ♦The analysis of infinitely repeated games helps to understand : – Cartels; – Self enforceable implicit agreements.
♦The stability of collusion.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
19 Trigger strategies in the waste management sector
In Dale County, in Florida, the waste management companies have acted colluded. The companies had private investigators following the other companies, and very low price contracts would be offered to every client of any deviating company.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
Mobile operators offer any contract of the competition
Either explicitly or at a client’s request, some mobile operators offer any contract of any competitor. This behavior, which looks like heavy competition, may really soften competition.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
The stability of collusion
♦Some factors affect collusion: – Frequency of transactions; – Number of competitors; – The dimension of the firms; – Market history.
Managerial Economics MBACatólica 2006-2007 • Fall Quarter • Session 7 ©Fernando Branco
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