Unit 4 Slides part 3
Strategy
A set of operating decisions based on the known choices that a player will make during the game. The purpose of the strategy is to adopt a plan that will lead to the maximum payoff!
The core of a cooperative game is:
An optimum and stable coalition structure
Pure Strategy
Complete list of choices of legal moves that a player will make for every possible situation that can occur during the game. An example is chess. If you can see all possible moves (as a computer can) then you make your move based on the best possible outcome.
John von Neumann & (1903 - 1957) Oskar Morgenstern (1902 - 1977)
In 1944 von Neumann and Morganstern published Theory of Games and Economic Behavior, the "Wealth of Nations" of game theory. The book moved game theory from the mathematics discipline to economics and other disciplines.
GAME: Stag Hunt
In Stag Hunt, a conclusion that Al and Bob could not trust each other to catch a stag left only one game solution: the non-cooperative (hare, hare) solution. But the game has another feature, and it is behavioral. By ranking the **(ordinal utility preferences)**, we will see that both players PREFER to catch stag, not hare.
Solution Set
In general, the set of all efficient (Pareto optimal) coalitions and payoffs that make every player at least as well off as he would be in a non-cooperative solution is called the solution set for the game. The solution set includes those solutions where total payoff is maximized.
Quantity-Driven Price- Flexibility Model
Now we change the model to a homogeneous product in a duopoly market. Both producers of computer paper (sold by the box) are permitted to manufacture the quantity of boxes they desire. In other words, there is now no assumption of fixed supply - in fact, the supply of boxes will increase!
When Do We Use Which Strategies?
Pure Strategies are a set of legal and rational strategies that contain various types of individual strategies, including dominant strategies and random strategies.
Three- to several-player game
The Queuing Game
WhydidVilfredoParetorelyon"moves" and "changes" to determine optimality?
Utility cannot be accurately measured
When to Adopt a Mixed Strategy
What happens when one player routinely "outsmarts" the other player at this game? In other words, what might a player who repeatedly loses pennies do in future rounds? The best option is to adopt a mixed strategy.
Analyzing Non-Cooperative Games
When analyzing ("breaking down") strategies of games, we typically try to convert cooperative games into non- cooperative games thereby making them true "strategic games". This also gives rise to the term "hybrid game" (which contains elements of both).
Side Payments
When part of the payoff is transferred from one member of a coalition to another, in order no member of the coalition needs to be worse off as a result of adopting the coordinated strategy of the coalition, this transfer is called a side payment.
Individual player payoffs in social welfare games:
do not matter
Nash Equilibrium
the Nash Equilibrium is the manner by which the players will choose their best strategy given the strategies that all the other players have chosen. In Prisoner's Dilemma, if The Beak refuses to confess then the best outcome for The Noggin is to also remain silent.
Social dilemmas can be identified by:
their ordinal utility preference profile
Economic efficiency is achieved when you eliminate:
unrealized potential
In game theory the spoiler can affect the outcome only:
when the winner was aided by the spoiler
Dominant Strategy
**(A strategy that is best for a player in a game regardless of the strategies chosen by the other player(s).)** In Prisoner's Dilemma the dominant strategy is to "confess" because it is the best strategy (lowest expected jail term) regardless of the choice made by the other player.
Dominant Strategy Equilibrium
**(The choice of a dominant strategy from a set of strategies that a player will select each time the game is played.)** For this to be true, a dominant strategy must exist and the payoff of the dominant strategy replayed over and over becomes the payoff of the dominant strategy equilibrium.
Types of Social Welfare Games
1. Constant Sum Games 2. Non-Constant Sum Games 3. Non-Zero Sum Games
Ways in Which Games Can Be Solved
1. EfficiencyTheories 2. UseofSaddlePoints 3. NashEquilibrium 4. ElusiveSolutions-MultipleNash Equilibria
Two Ways to Measure Market Success and Failure Under Game Theory
1. Maximin Theorem 2. Pareto Optimality
Game Theory History
1. ModernGameTheoryBegins1944 2. TheActivePeriod1945-55 3. Recent Game Theory Developments 4. SocialChoiceModels 5. PublicChoiceModels
Efficiency Theories to Solve Games
1. Pareto Optimality 2. Maximin Theory Both are solution theories for zero- sum games, however the maximin theory may apply to other games under certain conditions.
Some Strategies Worth Analyzing
1. Pure Strategy 2. Dominant Strategy 3. Random Strategy
Finding Nash Equilibria
1. Where do all players get at least "zero" payoff? 2. From that cell can any player switch strategies to maximize with a different payoff? Alternatively to searching for Nash Equilibria, determine if any coalition structures would be strong enough to prevent the existence of a core
Nash's Written Contributions
1950 - Devised the Nash Equilibrium to affirm solutions for non-cooperative games. 1950 - Extended bargaining theory by proving that a Nash Bargaining Solution exists as in game theory. 1951 - introduced the Nash Programme, whereby cooperative games were subject to reconstruction as strategic games. 1953 - Provided the first examples of the Nash Programme using two-person, cooperative games.
Expected Payoff
= (1 - Risk Factors ) * Present Value Where PV = Future Payoff / (1 + r) n, Where r = interest rate and n = number of years (or periods)
Coalition Structures
A division of the players in a game into coalitions (including singleton coalitions).
Monopolistic Competition
A market structure in which many firms sell products that are similar but not identical.
Oligopoly
A market structure in which only a few sellers offer similar or identical products.
Duopoly
A market structure in which only two sellers offer similar or identical products. Duopolies are a type of oligopoly. Oligopoly (duopoly in particular) is most obsessed with game theory. More game theory potential exists here than in other market structure types.
Finding Nash Equilibria
Alternatively to searching for Nash Equilibria, determine if any coalition structures would be strong enough to prevent the existence of a core
Law of Comparative Advantage
Although not directly tied to a specific market structure, game theory can have strong implications for trade in general, and international trade most significantly.
Backward Induction
An equilibrium refinement used to find Nash Equilibria in dynamic games with perfect information by solving subgames then substituting those solution payoffs back into earlier segments of the game tree.
Simultaneous and Sequential
An important distinction in how games function is in how the players move, and what information they have about the other player(s) at that time. Information availability matters to this classification of games just as it mattered to non-cooperative games.
Maximin Theorem
Another solution theory is the maximin theorem. The maximin is the strategy for which the minimum payoff for an action is the largest.
The Cournot Model
Assumption that an industry of two producers of a homogeneous good or service will view the difference between total output and the industry demand curve as excess demand to which each producer may supply, to which consequent downward price adjustments will result.
The Bertrand-Edgeworth Model
Based on the Cournot Model concept, this model proposed that the two duopolists would choose a price at which they could reasonably enter the market and profit. The quantity produced would be that which would support the price. Therefore, they avoid negative payoffs in disequilibrium.
Theory of Games and Economic Behavior
By including utility theory with game theory (second edition, 1947), the most significant result of the von Neumann-Morgenstern book was that the mathematical questions of the past arose as new economic challenges for the present.
References for Industry Demand Graph
C, TC = Cournot & "Total Cournot" M, TM = Monopoly & "Total Monopoly" X = Point where quantity rise starts and profit fall ends Y = Highest price at y-axis Z = Zero Profit point Total Quantity = Quantity of A + Quantity of B
Albert W. Tucker (1905 - 1995)
Canadian-born mathematician at Princeton University who collaborated with student Harold Kuhn and was the dissertation advisor to John Nash and Lloyd Shapley. Tucker is credited with applying the name of "Prisoner's Dilemma" to the Dresher and Flood experiments of 1950.
Random (or Mixed) Strategy
Choosing between two or more pure strategies in a normal form game is known as random or mixed strategy. Players will switch to mixed strategies only when the expected or average payoff is at least as large as that obtained by any other strategy, and a mixed strategy is part of the set of pure strategies available to the player. This is illustrated in Matching Pennies.
David Ricardo (1772 - 1823) `
Classical economist who espoused the Law of Comparative Advantage postulating **(that two nations are always better off when they produce the one of two goods in which they better specialize)** (i.e., the "comparative advantage"). The excess of the good should be exported to the trading partner in exchange for the good in which it specializes.
Coalition Dominance
Coalition dominance refers to the ability of a coalition to provide higher payoffs to its members than the members would receive in another coalition or individually.
Pareto Optimality
Condition that exists in a social organization when no change can be implemented that will make someone better off without making someone else worse off - each in his or her own estimation.
Analyzing Cooperative Games
Cooperative games (coalitional games) work like "Big Brother" on TV. Players do what they can by enlisting the help of others to move them toward the payoff.
Cooperative and Non-Cooperative
Cooperative games are also known as "coalitional games". Non-cooperative games are also known as "strategic games". Hybrid games are both, and may contain coalitional players who act as individuals outside the coalition.
Information Availability
Developing strategies and determining which strategy to play hinges on the information set available to the player. In some games, perfect information is known (such as chess), but not always. Strategy choices are dependent on what information is known at the time.
Harold Hotelling
Economist and statistician Harold Hotelling studied the behavior of monopolistically competitive firms. In 1929 he developed the Hotelling Paradox to explain the actions of these firms in the product market.
Merrill M. Flood & Melvin Dresher (1912 - 1991) (1911 - 1992)
Flood left Princeton University for RAND Corporation and headed a 1950 project with Dresher called "the Flood-Dresher Experiments". The experiments were run multiple times using subjects who had prescribed payoffs and two strategies. Results were published by Flood in 1952.
Augustin Cournot (1801 - 1877)
French mathematician and philosopher whose 1838 publication Recherches first explained how a duopolist would increase quantity based on the rival's expected output, which was always less than actual output, leading to overproduction and lower prices.
Zero Sum Game
Game in which the payoffs for the players always add up to zero.
Centipede Game
Game that allows players to choose between small short-term payoffs or larger long-term payoffs. Re- plays of the game shows that players do not consistently follow equilibrium strategies.
Harold W. Kuhn (1925 - )
Game theory mathematician at Princeton University. Always a close associate of John Nash, Kuhn was co-editor of the book The Essential John Nash which popularized Nash leading to his selection for the Nobel prize and the book and movie A Beautiful Mind, in which Kuhn served as advisor.
Non-Constant Sum Game
Game where the sum of the payoffs add up to a different constant for any combination of strategies selected by the players.
Constant Sum Game
Game where the sum of the payoffs will always be the same regardless of the combination of strategies selected by the players.
Simultaneous Games
Games in which players move simultaneously or at least without knowing what other player(s) moves will be (normal form).
Sequential Games
Games in which players take turns and each move (strategy decision) is known to the other player(s) thereby allowing it to be calculated into the information set.
What is a Price War?
In this case we are creating a "war" between two sellers of identical or substitutable goods (high-end clothing). Both stores have a limited quantity of the clothing they must sell before the season's end. The lower prices are intended to exhaust the supply.
Path to Pareto Optimality
Individuals determine their own utility ("likes" and "dislikes") External observers cannot presume to make comparisons of utility among separate individuals - only the individuals themselves can do that.
Transferable Utility (TU)
Interpersonal standardized consideration added to one player's wealth and subtracted from the wealth of another player or players.
Infinite player number game
Lotto
Two-player game
Matching Pennies
Nash Equilibrium
Named for the mathematician John Nash,**(the Nash Equilibrium is the manner by which the players will choose their best strategy given the strategies that all the other players have chosen.)** If The Beak refuses to confess then the best outcome for The Noggin is to also remain silent.
John Nash (1928 - )
Nobel prize laureate (1994) whose major contributions began while still a graduate student at Princeton. His four significant papers were extensions of problems presented in von Neumann- Morgenstern's book, which had been revised to include utility in 1947.
Non-Constant Sum Games as a Class
Non-constant games are also referred to as variable constant games. Non-zero sum games MAY be members of this class, depending on if the sum of the payoffs varies across the combination of strategies.
Employing Game Theory
Note that an important distinction between the dominant strategy and the Nash Equilibrium depends on the rules of the game (i.e., is cooperation permitted?). Knowing the different outcomes, apply what you know about game theory to oligopoly market structures. Now, why do you think collusion is illegal?
Coalition Dynamics
Players in coalitions can move their coalitions just as individual players do in non-cooperative games, however coalitions add another dimension: a player may join or leave a coalition when the expected payoffs dictate that such a move is rational.
Hotelling Paradox
Proposition that states that monopolistically competitive firms must, in order to attract each other's customers, make their products as similar to existing products as possible without destroying the differences. Therefore, as a type of market structure, monopolistic competition leads to maximum differentiation of products with minimum differences between them.
Effect of Quantity on Profits
So if price goes down as quantity is expanded, what effect does this have on the profits of the two duopolists? In other words, do profits go up or down?
Games Grouped by Social Welfare
Social Welfare is concerned with the overall benefits and costs that accrue to all of the players, regardless of how they those benefits and costs are spread among individual players.
Social Dilemmas and Ordinal Numbers
Social dilemmas such as Stag Hunt, and others such as the Dumping Game and Heave-Ho, can be shown as the same in strict ordinal terms because the strategy rankings for these games fit similar profiles. Check Prisoner's Dilemma's payoffs - you see the same pattern. So what is the difference? Prisoner's Dilemma is still a trust dilemma, **(but it is a non-cooperative game!)**
Strategy and Information Sets
Strategies are driven by the information set (what is and is not known to the player at the time). Under sequential games we will explore further the impact "perfect information" has on game theory analysis.
Coalitions
The Essence of Cooperative Games
The first written game that took over 200 years to solve was:
The Montmort minimax game
The notion that any cooperative game can be reconstituted as a strategic game is:
The Nash Programme (developed by John Nash)
The year1944 is significant to the history of game theory because in 1944:
The Theory of Games and Economic Behavior was published
Constant Sum Games as a Class
The Zero-sum property means one must lose for the other to gain. This is a (Pareto optimal solution). It is equivalent to deciding how big to cut the pieces of a pie, but the size of the pie cannot be changed.
The Core
The core of a cooperative game consists of all coalition structures (if there are any) that are stable in the sense that there is no individual or group that can improve their payoffs (including side payments) by dropping out or reorganizing to form a new or separate coalition.
Pareto Optimality (Definition)
The criterion that social welfare economics employs in determining whether or not a given situation is "efficient" or "optimal" and whether or not a given move or change is "efficient" or "optimal".
What is a coalition structure?
The division of players into coalitions
Prisoner's Dilemma
The essence of Prisoner's Dilemma is that two "prisoners" each have two choices. Pursuit of their own self-interest (which is confession, to receive the lightest prison sentence) yields a worse alternative (10 years in prison).**(The game illustrates why cooperation is difficult to maintain even when it is mutually beneficial.)***
Saddle Point
The saddle point refers to when the maximin and the minimax are identical. It is therefore considered the ideal solution to a zero-sum game solved by the Maximin Principle.
"Commitment structure" is:
The set of all subgames
Dominant Strategy
The strategy a player selects because it yields higher payoffs than all other available strategies. Based on known expected payoffs in the game, players will typically choose a dominant strategy (if one exists) at the outset.
GAME: Three Farmers' Road
This combines public choice theory with game theory. Three farmers are considering a road paving project for their dirt road. All are self- interested seeking to maximize benefits while minimizing costs, i.e., wanting to pave the road only to their farm. We can view this in the form of possible solution sets, which are the strategies: i (individual), ii (two-farmer coalition), and iii (unanimity).
GAME: Prisoner's Dilemma
Two captured accomplices, Jimmy "The Noggin" and Johnny "The Beak". Police know these two are guilty, but don't have all the evidence to win a long-term conviction. How do they put them away for a long time?
Prisoner's Dilemma
Without all the necessary evidence, a good outcome for the police is to get one of the suspects to confess thereby sending the other to prison for 25 years. If both confess, then they both go to prison for 10 years, also a good outcome. Only the "remain silent/remain silent" option is not desirable.
Constant Sum Games as a Class
Zero Sum Games are a special type of constant sum games. Non-Zero Sum Games only mean that the sum of the payoffs is NOT ZERO (regardless of whether it is constant)
Which statement is true only when you have achieved Pareto optimality?
any utility gain means a player lost equal utility
Symmetric and Asymmetric Games
are similar to social welfare games in that the secret to their power is held in the payoffs, but here the difference is in whether the payoffs are "fair" from the vantage of players of the game.
A game is a
competitive and risk-taking venture with prescribed rules that must be navigated by any number of players who, through their own decisions and moves, seek expected payoffs as their reward for playing.
One way to expand the solution set is to:
create side payments
In the Queuing game, players choose to stand or sit based on:
expected payoffs
When the expected payoff is equal to or more than that of the current strategy, the player may be switching:
from a dominant strategy to a mixed strategy
Games that allow side payments require the use of:
interpersonal standardized considerations
Game theory
is the application of statistics and rational decision-making to goal-oriented strategic situations of competition and cooperation while operating under specified rules.
Using the Maximin Principle, a player's goal is to:
maximize the minimum payoff
