NationMaster - Encyclopedia: Stackelberg competition

The Stackelberg leadership model is a strategic game in economics in which the leader firm moves first and then the follower firms move sequentially. It is named after the German economist Heinrich Freiherr von Stackelberg who published Marktform und Gleichgewicht in 1934 which described the model. Face-to-face trading interactions on the New York Stock Exchange trading floor. ... Heinrich Freiherr von Stackelberg (1905-1946) is an economist who contributed to game theory. ... Year 1934 (MCMXXXIV) was a common year starting on Monday (link will display full 1934 calendar) of the Gregorian calendar. ...

In game theory terms, the players of this game are a leader and a follower and they compete on quantity. The Stackelberg leader is sometimes referred to as the Market Leader. Game theory is a branch of applied mathematics that is often used in the context of economics. ...

There are some further constraints upon the sustaining of a Stackelberg equilibrium. The leader must know ex ante that the follower observes his action. The follower must have no means of committing to a future non-Stackelberg follower action and the leader must know this. Indeed, if the 'follower' could commit to a Stackelberg leader action and the 'leader' knew this, the leader's best response would be to play a Stackelberg follower action. Ex ante is a Latin term meaning beforehand. Ex ante evaluations deal with forecasting and forecasted returns on invested money. ...

Firms may engage in Stackelberg competition if one has some sort of advantage enabling it to move first. More generally, the leader must have commitment power. Moving observably first is the most obvious means of commitment: once the leader has made its move, it cannot undo it - it is committed to that action. Moving first may be possible if the leader was the incumbent monopoly of the industry and the follower is a new entrant. Holding excess capacity is another means of commitment.

1 Subgame perfect Nash equilibrium
- 1.1 Examples
2 Economic analysis
3 Noncredible threats by the follower
4 Stackelberg compared with Cournot
- 4.1 Game theoretic considerations
5 Comparison with other oligopoly models
6 See also
7 References

Subgame perfect Nash equilibrium

The Stackelberg model can be solved to find the subgame perfect Nash equilibrium or equilibria (SPNE), i.e. the strategy profile that serves best each player, given the strategies of the other player and that entails every player playing in a Nash equilibrium in every subgame. Subgame perfect equilibrium is an economics term used in game theory to describe an equilibrium such that players strategies constitute a Nash equilibrium in every subgame of the original game. ... Subgame perfect equilibrium is an economics term used in game theory to describe an equilibrium such that players strategies constitute a Nash equilibrium in every subgame of the original game. ... In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. ... A minigame is a (usually short) segment of a video game that uses a different style of gameplay than the rest of the game. ...

In very general terms, let the price function for the (duopoly) industry be $P (q 1 + q 2)$ where the subscript 1 represents the leader and 2 represents the follower. Price is simply a function of total (industry) output. Suppose firm i has the cost structure $C i (q i)$ . The model is solved by backward induction. The leader considers what the best response of the follower is, i.e. how it will respond once it has observed the quantity of the leader. The leader then picks a quantity that maximizes its payoff, anticipating the predicted response of the follower. The follower actually observes this and in equilibrium picks the expected quantity as a response. In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... In game theory, the best response, is the strategy (or strategies) which produces the most favorable immediate outcome for the current player, taking other players strategies as given. ...

To calculate the SPNE, the best response functions of the follower must first be calculated (calculation moves 'backwards' because of backward induction). In game theory, the best response, is the strategy (or strategies) which produces the most favorable immediate outcome for the current player, taking other players strategies as given. ...

The profit of firm 2 (the follower) is revenue less cost. Revenue is the product of price and quantity and cost is given by the firm's cost structure, so profit is: $Pi_2 = P(q_1+q_2) cdot q_2 - C_2(q_2)$ . The best response is to find the value of $q 2$ that maximises $Π 2$ given $q 1$ , i.e. given the output of the leader (firm 1), the output that maximises the follower's profit is found. Hence, the maximum of $Π 2$ with respect to $q 2$ is to be found. First derive $Π 2$ with respect to $q 2$ :

$frac{partial Pi_2 }{partial q_2} = frac{partial P(q_1+q_2) }{partial q_2} cdot q_2 + P(q_1+q_2) - frac{partial C_2 (q_2)}{partial q_2}.$

Setting this to zero for maximization:

$frac{partial Pi_2 }{partial q_2} = frac{partial P(q_1+q_2) }{partial q_2} cdot q_2 + P(q_1+q_2) - frac{partial C_2 (q_2)}{partial q_2}=0.$

The values of $q 2$ that satisfy this equation are the best responses. Now the best response function of the leader is considered. This function is calculated by considering the follower's output as a function of the leader's output, as just computed.

The profit of firm 1 (the leader) is $Π 1 = P (q 1 + q 2 (q 1)). q 1 - C 1 (q 1)$ , where $q 2 (q 1)$ is the follower's quantity as a function of quantity, namely the function calculated above. The best response is to find the value of $q 1$ that maximises $Π 1$ given $q 2 (q 1)$ , i.e. given the best response function of the follower (firm 2), the output that maximises the leader's profit is found. Hence, the maximum of $Π 1$ with respect to $q 1$ is to be found. First derive $Π 1$ with respect to $q 1$ :

$frac{partial Pi_1}{partial q_1} = frac{partial P(q_1+q_2)}{partial q_2} cdot frac{partial q_2(q_1)}{partial q_1} cdot q_1 + P(q_1+q_2(q_1)) - frac{partial C_1 (q_1)}{partial q_1}.$

Setting this to zero for maximization:

$frac{partial Pi_1}{partial q_1} = frac{partial P(q_1+q_2)}{partial q_2} cdot frac{partial q_2(q_1)}{partial q_1} cdot q_1 + P(q_1+q_2(q_1)) - frac{partial C_1 (q_1)}{partial q_1}=0.$

Examples

The following example is very general. It assumes a generalised linear demand structure and imposes some restrictions on cost structures for simplicity's sake so the problem can be resolved.

${partial ^2C_i (q_i)}{partial q_i cdot partial q_j}=0,forall j$ and $frac{partial C_i (q_i)}{partial q_j}=0, j ne i$

for ease of computation.

The follower's profit is:

$Pi_2 = bigg(a - b(q_1+q_2)bigg) cdot q_2 - C_2(q_2).$

The maximisation problem resolves to (from the general case):

$frac{partial bigg(a - b(q_1+q_2)bigg) }{partial q_2} cdot q_2 + a - b(q_1+q_2) - frac{partial C_2 (q_2)}{partial q_2}=0,$

$Rightarrow - bq_2 + a - b(q_1+q_2) - frac{partial C_2 (q_2)}{partial q_1}=0,$

$Rightarrow q_2 = frac{a - bq_1 - frac{partial C_2 (q_2)}{partial q_2}}{2b}.$

Consider the leader's problem:

$Pi_1 = bigg(a - b(q_1+q_2(q_1))bigg) cdot q_1 - C_1(q_1).$

Substituting for $q 2 (q 1)$ from the follower's problem:

$Pi_1 = bigg(a - bbigg(q_1+frac{a - bq_1 - frac{partial C_2 (q_2)}{partial q_2}}{2b}bigg)bigg) cdot q_1 - C_1(q_1),$

$Rightarrow Pi_1 = bigg(frac{a - b.q_1 + frac{partial C_2 (q_2)}{partial q_2}}{2})bigg) cdot q_1 - C_1(q_1).$

The maximisation problem resolves to (from the general case):

$frac {partial Pi_1}{partial q_1} = bigg(frac{a - 2bq_1 + frac{partial C_2 (q_2)}{partial q_2}}{2}bigg) - frac{partial C_1 (q_1)}{partial q_1}=0.$

Now solving for $q 1$ yields $q_1^*$ , the leader's optimal action:

$q_1^*=frac{a + frac{partial C_2 (q_2)}{partial q_2}- 2 cdot frac{partial C_1 (q_1)}{partial q_1}}{2b}.$

This is the leader's best response to the reaction of the follower in equilibrium. The follower's actual can now be found by feeding this into its reaction function calculated earlier:

$q_2^* = frac{a - b cdot frac{a + frac{partial C_2 (q_2)}{partial q_2}- 2 cdot frac{partial C_1 (q_1)}{partial q_1}}{2b} - frac{partial C_2 (q_2)}{partial q_2}}{2b},$

$Rightarrow q_2^* = frac{a - 3 cdot frac{partial C_2 (q_2)}{partial q_2}+ 2 cdot frac{partial C_1 (q_1)}{partial q_1}}{4b}.$

The Nash equilibria are all $(q_1^*, q_2^*)$ . It is clear (if marginal costs are assumed to be zero - i.e. cost is essentially ignored) that the leader has a significant advantage. Intuitively, if the leader was no better off than the follower, it would simply adopt a Cournot competition strategy. Cournot competition is an economic model used to describe industry structure. ...

Economic analysis

An extensive-form representation is often used to analyze the Stackelberg leader-follower model. Also referred to as a “decision tree”, the model shows the combination of outputs and payoffs both firms have in the Stackelberg game In operations research, specifically in decision analysis, a decision tree is a decision support tool that uses a graph or model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. ...

A Stackelberg game represented in extensive form

The image on the left depicts in extensive form a Stackelberg game. The payoffs are shown on the right. This example is fairly simple. There is a basic cost structure involving only marginal cost (there is no fixed cost). The demand function is linear and price elasticity of demand is 1. However, it illustrates the leader's advantage. Image File history File links Download high resolution version (962x423, 33 KB)Extensive form game 1A Treborbassett 00:13, 14 Mar 2005 (UTC) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Download high resolution version (962x423, 33 KB)Extensive form game 1A Treborbassett 00:13, 14 Mar 2005 (UTC) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... It has been suggested that Game tree be merged into this article or section. ... It has been suggested that Game tree be merged into this article or section. ... In economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit. ... Fixed costs are expenses whose total does not change in proportion to the activity of a business, within the relevant time period or scale of production. ...

The follower wants to choose $q 2$ to maximise its payoff $5000 - q 1 - q 2 - c 2$ . Taking the first order derivative and equating it to zero (for maximisation) yields $q_2=frac{5000-q_1-c_2}{2}$ as the maximum value of $q 2$ .

The leader wants to choose $q 1$ to maximise its payoff $5000 - q 1 - q 2 - c 1$ . However, in equilibrium, it knows the follower will choose $q 2$ as above. So in fact the leader wants to maximise its payoff $5000-q_1-frac{5000-q_1-c_2}{2}-c_1$ (by substituting $q 2$ for the follower's best response function). By differentiation, the maximum payoff is given by $q_1=frac{5000-2c_1+c_2}{2}$ . Feeding this into the follower's best response function yields $q_2=frac{5000+2c_1-3c_2}{4}$ . Suppose marginal costs were equal for the firms (so the leader has no market advantage other than first move) and in particular $c 1 = c 2 = 1000$ . The leader would produce 2000 and the follower would produce 1000. This would give the leader a profit (payoff) of two million and the follower a profit of one million. Simply by moving first, the leader has accrued twice the profit of the follower. However, Cournot profits here are 1.78 million apiece (strictly, $(16 / 9)10 6$ apiece), so the leader has not gained much, but the follower has lost. However, this is example-specific. There may be cases where a Stackelberg leader has huge gains beyond Cournot profit that approach monopoly profits (for example, if the leader also had a large cost structure advantage, perhaps due to a better production function). There may also be cases where the follower actually enjoys higher profits than the leader, but only because it, say, has much lower costs. Cournot competition is an economic model used to describe industry structure. ... This article is about the economics of markets dominated by a single seller. ... In microeconomics, a production function expresses the relationship between an organizations inputs and its outputs. ...

Noncredible threats by the follower

If, after the leader had selected its equilibrium quantity, the follower deviated from the equilibrium and chose some non-optimal quantity it would not only hurt itself, but it could also hurt the leader. If the follower chose a much larger quantity than its best response, the market price would lower and the leader's profits would be stung, perhaps below Cournot level profits. In this case, the follower could announce to the leader before the game starts that unless the leader chooses a Cournot equilibrium quantity, the follower will choose a deviant quantity that will hit the leader's profits. After all, the quantity chosen by the leader in equilibrium is only optimal if the follower also plays in equilibrium. The leader is, however, in no danger. Once the leader has chosen its equilibrium quantity, it would be irrational for the follower to deviate because it too would be hurt. Once the leader has chosen, the follower is better off by playing on the equilibrium path. Hence, such a threat by the follower would be incredible.

However, in an (indefinitely) repeated Stackelberg game, the follower might adopt a punishment strategy where it threatens to punish the leader in the next period unless it chooses a non-optimal strategy in the current period. This threat is credible because it would be rational for the follower to punish in the next period so that the leader chooses Cournot quantities thereafter.

Stackelberg compared with Cournot

The Stackelberg and Cournot models are similar because in both competition is on quantity. However, as seen, the first move gives the leader in Stackelberg a crucial advantage. There is also the important assumption of perfect information in the Stackelberg game: the follower must observe the quantity chosen by the leader, otherwise the game reduces to Cournot. With imperfect information, the threats described above can be credible. If the follower cannot observe the leader's move, it is no longer irrational for the follower to choose, say, a Cournot level of quantity (in fact, that is the equilibrium action). However, it must be that there is imperfect information and the follower is unable to observe the leader's move because it is irrational for the follower not to observe if it can once the leader has moved. If it can observe, it will so that it can make the optimal decision. Any threat by the follower claiming that it will not observe even if it can is as uncredible as those above. This is an example of too much information hurting a player. In Cournot competition, it is the simultaneity of the game (the imperfection of knowledge) that results in neither player (ceteris paribus) being at a disadvantage. Cournot competition is an economic model used to describe industry structure. ... Perfect information is a term used in economics and game theory to describe a state of complete knowledge about the actions of other players that is instantaneously updated as new information arises. ... Ceteris paribus is a Latin phrase, literally translated as with other things [being] the same, and usually rendered in English as all other things being equal. ...

Game theoretic considerations

As mentioned, imperfect information in a leadership game reduces to Cournot competition. However, some Cournot strategy profiles are sustained as Nash equilibria but can be eliminated as incredible threats (as described above) by applying the solution concept of subgame perfection. Indeed, it is the very thing that makes a Cournot strategy profile a Nash equilibrium in a Stackelberg game that prevents it from being subgame perfect. Game theory is a branch of applied mathematics that is often used in the context of economics. ... In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ... In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... Subgame perfect equilibrium is an economics term used in game theory to describe an equilibrium such that players strategies constitute a Nash equilibrium in every subgame of the original game. ...

Consider a Stackelberg game (i.e. one which fulfills the requirements described above for sustaining a Stackelberg equilibrium) in which, for some reason, the leader believes that whatever action it takes, the follower will choose a Cournot quantity (perhaps the leader believes that the follower is irrational). If the leader played a Stackelberg action, (it believes) that the follower will play Cournot. Hence it is non-optimal for the leader to play Stackelberg. In fact, its best response (by the definition of Cournot equilibria) is to play Cournot quantity. Once it has done this, the best response of the follower is to play Cournot.

Consider the following strategy profiles: the leader plays Cournot; the follower plays Cournot if the leader plays Cournot and the follower plays non-Stackelberg if the leader plays Stackelberg and if the leader plays something else, the follower plays an arbitrary strategy (hence this actually describes several profiles). This profile is a Nash equilibrium. As argued above, on the equilibrium path play is a best response to a best response. However, playing Cournot would not have been the best response of the leader were it that the follower would play Stackelberg if it (the leader) played Stackelberg. In this case, the best response of the leader would be to play Stackelberg. Hence, what makes this profile (or rather, these profiles) a Nash equilibrium (or rather, Nash equilibria) is the fact that the follower would play non-Stackelberg if the leader were to play Stackelberg.

However, this very fact (that the follower would play non-Stackelberg if the leader were to play Stackelberg) means that this profile is not a Nash equilibrium of the subgame starting when the leader has already played Stackelberg (a subgame off the equilibrium path). If the leader has already played Stackelberg, the best response of the follower is to play Stackelberg (and therefore it is the only action that yields a Nash equilibrium in this subgame). Hence the strategy profile - which is Cournot - is not subgame perfect.

Comparison with other oligopoly models

In comparison with other oligopoly models,

The aggregate Stackelberg output is greater than the aggregate Cournot output, but less than the aggregate Bertrand output.

The Stackelberg price is lower than the Cournot price, but greater than the Bertrand price.

The Stackelberg consumer surplus is greater than the Cournot consumer surplus, but lower than the Bertrand consumer surplus.

The aggregate Stackelberg output is greater than pure monopoly or cartel, but less than the perfectly competitive output.

The Stackelberg price is lower than the pure monopoly or cartel price, but greater than the perfectly competitive price.

Cournot competition is an economic model used to describe industry structure. ... Bertrand competition is a model of competition used in economics, named after Joseph Louis FranÃ§ois Bertrand (1822-1900). ... This article is about the economics of markets dominated by a single seller. ... For the American pop-punk band, see Cartel (band). ... Competition is the act of striving against another force for the purpose of achieving dominance or attaining a reward or goal, or out of a biological imperative such as survival. ...

References

Fudenberg, D. and Tirole, J. (1993) Game Theory, MIT Press. (see Chapter 3, sect 1)
Gibbons, R. (1992) A primer in game theory, Harvester-Wheatsheaf. (see Chapter 2, section 1B)
Osborne, M.J. and Rubenstein, A. (1994) A Course in Game Theory, MIT Press (see p 97-98)

Jean Tirole (born 9 August 1953) is a notable contemporary french economist, author of many works in economics, scientific director of the Industrial Economics Institute in Toulouse. ... Ariel Rubinstein (born April 13, 1951) is an economist who works in game theory. ...

view	Topics in game theory
Definitions Game theory is a branch of applied mathematics that is often used in the context of economics. ...	Normal form game · Extensive form game · Cooperative game · Information set · Preference In game theory, normal form is a way of describing a game. ... It has been suggested that Game tree be merged into this article or section. ... A cooperative game is a game where groups of players (coalitions) may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players. ... In game theory, an information set is a set that, for a particular player, establishes all the possible moves that could have taken place in the game so far, given what that player has observed so far. ... Preference (or taste) is a concept, used in the social sciences, particularly economics. ...
Equilibrium concepts Price of market balance In economics, economic equilibrium is simply a state of the world where economic forces are balanced and in the abscence of external shocks the (equilibrium) values of economic variables will not change. ... In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ...	Nash equilibrium · Subgame perfection · Bayesian-Nash · Perfect Bayesian · Trembling hand · Proper equilibrium · Epsilon-equilibrium · Correlated equilibrium · Sequential equilibrium · Quasi-perfect equilibrium · Evolutionarily stable strategy · Risk dominance In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. ... Subgame perfect equilibrium is an economics term used in game theory to describe an equilibrium such that players strategies constitute a Nash equilibrium in every subgame of the original game. ... In game theory, a Bayesian game is one in which information about characteristics of the other players (i. ... In game theory, a Bayesian game is one in which information about characteristics of the other players (i. ... The trembling hand perfection is a notion that eliminates actions of players that are unsafe because they were chosen through a slip of the hand. ... Proper equilibrium is a refinement of Nash Equilibrium due to Roger B. Myerson. ... In game theory, an Epsilon-equilibrium is a strategy profile that approximately satisfies the condition of Nash Equilibrium. ... In game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. ... Sequential equilibrium is a refinement of Nash Equilibrium for extensive form games due to David M. Kreps and Robert Wilson. ... Quasi-perfect equilibrium is a refinement of Nash Equilibrium for extensive form games due to Eric van Damme. ... In game theory, an evolutionarily stable strategy (or ESS; also evolutionary stable strategy) is a strategy which if adopted by a population cannot be invaded by any competing alternative strategy. ... Risk dominance and payoff dominance are two related refinements of the Nash equilibrium (NE) solution concept in game theory, defined by John Harsanyi and Reinhard Selten. ...
Strategies In game theory, a players strategy, in a game or a business situation, is a complete plan of action for whatever situation might arise; this fully determines the players behaviour. ...	Dominant strategies · Mixed strategy · Tit for tat · Grim trigger · Collusion In game theory, dominance occurs when one strategy is better or worse than another regardless of the strategies of a players opponents. ... In game theory a mixed strategy is a strategy which chooses randomly between possible moves. ... Tit for Tat is a highly-effective strategy in game theory for the iterated prisoners dilemma. ... Grim Trigger is a trigger strategy in game theory for a repeated game, such as an iterated prisoners dilemma. ... Look up collusion in Wiktionary, the free dictionary. ...
Classes of games	Symmetric game · Perfect information · Dynamic game · Repeated game · Signaling game · Cheap talk · Zero-sum game · Mechanism design · Stochastic game · Nontransitive game In game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. ... Perfect information is a term used in economics and game theory to describe a state of complete knowledge about the actions of other players that is instantaneously updated as new information arises. ... In game theory, a sequential game is a game where one player chooses his action before the others chooses theirs. ... In game theory, a repeated game (or iterated game) is an extensive form game which consists in some number of repetitions of some base game (called a stage game). ... Signaling games are dynamic games with two players, the sender (S) and the receiver (R). ... Cheap Talk is a term used in Game Theory for pre-play communication which carries no cost. ... Zero-sum describes a situation in which a participants gain (or loss) is exactly balanced by the losses (or gains) of the other participant(s). ... Mechanism design is a sub-field of game theory. ... In game theory, a stochastic game is a competitive game with probabilistic transitions played by two players. ... A non-transitive game is a game for which the various strategies produce one or more loops of preferences. ...
Games Game theory studies strategic interaction between individuals in situations called games. ...	Prisoner's dilemma · Traveler's dilemma · Coordination game · Chicken · Volunteer's dilemma · Dollar auction · Battle of the sexes · Stag hunt · Matching pennies · Ultimatum game · Minority game · Rock, Paper, Scissors · Pirate game · Dictator game · Public goods game · Nash bargaining game · Blotto games · War of attrition Will the two prisoners cooperate to minimize total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? In game theory, the prisoners dilemma (sometimes abbreviated PD) is a type of non-zero-sum game in which two players... In game theory, the travelers dilemma (sometimes abbreviated TD) is a type of non-zero-sum game in which two players attempt to maximise their own payoff, without any concern for the other players payoff. ... In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ... For other uses, see Chicken (disambiguation). ... The Volunteers dilemma game models a situation in which each of N players faces the decision of either making a small sacrifice from which all will benefit or freeriding. ... On eBay, where an auction has a starting price of $1 ... The Battle of the Sexes is a two player game used in game theory. ... In game theory, the Stag Hunt is a game first discussed by Jean-Jacques Rousseau. ... Matching Pennies is the name for a simple example game used in game theory. ... The Ultimatum game is an experimental economics game in which two parties interact anonymously and only once, so reciprocation is not an issue. ... Minority Game is a game proposed by Yi-Cheng Zhang and Damien Challet from the University of Fribourg. ... Rock, Paper, Scissors chart Listen to this article ( info/dl) This audio file was created from an article revision dated 2006-07-13, and may not reflect subsequent edits to the article. ... From Howard Pyles Book of Pirates The pirate game is a simple mathematical game. ... The dictator game is a very simple game in experimental economics, similar to the ultimatum game. ... The Public goods game is a standard of experimental economics; in the basic game subjects secretly choose how many of their private tokens to put into the public pot. ... The Nash Bargaining Game is a simple two player game used to model bargaining interactions. ... Blotto games (or Colonel Blotto games) constitute a class of two-person zero-sum games in which the players are tasked to simultaneously distribute limited resources over several objects, with the gain (or payoff) being equal to the sum of the gains on the individual objects. ... In game theory the War of attrition is a model of aggression in which two contestants compete for a resource of value V by persisting while accumulating costs at a constant rate c. ...
Theorems	Minimax theorem · Purification theorems · Folk theorem · Revelation principle · Arrow's theorem â€œMinmaxâ€ redirects here. ... In game theory, the purification theorem was contributed by Nobel laurate John Harsanyi in 1973[1]. The theorem aims to justify a puzzling aspect of mixed strategy Nash equilibria: that each player is wholly indifferent amongst each of the actions he puts non-zero weight on, yet he mixes them... In game theory, folk theorems are a class of theorems which imply that in repeated games, any outcome is a feasible solution concept, if under that outcome the players minimax conditions are satisfied. ... The revelation principle of economics can be stated as, To any equilibrium of a game of incomplete information, there corresponds an associated revelation mechanism that has an equilibrium where the players truthfully report their types. ... In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox demonstrates the impossibility of designing a set of rules for social decision making that would meet all of a certain set of criteria. ...

Categories: Economics models | Game theory | Competition

Results from FactBites:

NodeWorks - Encyclopedia: Stackelberg competition (1834 words)
	The Stackelberg leadership model is a model of duopoly in economics.
	There may be cases where a Stackelberg leader has huge gains beyond Cournot profit that approach monopoly profits (for example, if the leader also had a large cost structure advantage, perhaps due to a better production function).
	However, in an (indefinitely) repeated Stackelberg game, the follower might adopt a punishment strategy where it threatens to punish the leader in the next period unless it chooses a non-optimal strategy in the current period.

Intertic, International Think-Tank on Innovation and Competition (448 words)
	Federico Etro is Associate Professor of Economics at the University of Milan (Department of Economics), where he teaches Microeconomics and Competition Policy, and Director of ECG, a small company for economic and industrial consulting based in Milan.
	His main research is about the economic theory of the role of market leaders in alternative competitive frameworks.
	Before obtaining tenured position in the Department of Economics of the University of Milan, he has taught at the Graduate School in Public Economics of Milan and Luiss University in Rome, and has been Teaching Fellow for graduate courses at the Kennedy School of Government of Harvard University.

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