VWL GT

• An action which is never a best response, no matter which action the other players chooses OR 
• an action with a payoff that is always lower than the payoff of another action, no matter which action the other players choose. 

an action that is always better(compared to another action) for one player , no matter what the other player chooses, since it always provides a higher payoff that 

an action which is at least as preferred as any other action no matter what the other player does but at least for one action that the other player chooses it must be strictly better. OR 

an action which has a payoff which is ...

MERKE: at least as good as other action and sometimes better (mind einmal) 

a game where one players loss is another players win. zB matching pennies, rock paper scissors

Spaniell: a game in which the sum of all players' payoffs equals 0 for every outcome. Also in jedem Kastel summieren und das muss 0 ergeben!

a function which assigns a certain payoff to a player at a certain action or strategy profile:

U1 (Quiet,Fink3) = 3

U1(a,a)=efef

erg -->Y stichwort einpendeln und spannungszustand

1. a situation in which no player can gain if he chooses another action (than a*), given that the other players choose action a-i* (weil payoffs hier am höchsten)
2. a*, given that the other players choose a* is at least as good as any other action, given that the other players choose a* --> mathematisch überlegen (wieder weil pay
3. the payoff of action a* is higher or at least as high as the payoff of any other action, given that the other players choose a* 
4. An action profile in which every players action is a best response to the other players action

Indifferenz gegenüber actions, weil payoffs gleich hoch sind, Daher auch keine Abweichung von dieser Siitation erwünscht von einem rational player. 

• a steady state in which no player wants to deviate (change his or her action/strategy), given the that the other player chooses a-i* or s-i* (the action/strategy in NE), since this action/strategy provides (If the other player chooses a*/s*) (expected) payoffs that are at least as high as if he would choose another action or strategy. 
• every player plays a best response to the other players action/strategy --> player 1s action is a best response to player 2s action and vice versa. 

A best response function represents a set of actions, with the property that the payoff of this set of actions is at least as large as the payoff to any other action. Die payoff funktion repräsentiert alle actions von player i, die eine "beste Reaktion" iSv highest payoffs auf die action. --> unsere payoffs sind also maximal, daher müssen wir auch unsere utility function ableiten...

zB B1 (Fink) = {Fink} oder B1(M) = {L,C}

no 

paretio effizient: An outcome is pareto optimal if there is no outcome that makes at least one player better off without making any player worse off (Spaniell "comparing outcomes and pareto omptimality) 

keiner kann mehr besser gestellt werden, ohne dass der andere schlechter gestellt wird! 

Gegenteil: ineffizient

Something that one player does to convince the other player, that he will choose a certain action. Such a strategic commitment must be: 

• visible and understandable for the other player(s)
• irreversible or at least hard to reverse (zB Zwangsjacke anziehen bei Chicken game, Investition tätigen bei market entry game, sodass man dominant action erhält! 

--> dadurch wird quasi sequential move game kreeirt. The other person doesnt move first, but creates the same 

A pure strategy defines an action or strategy that a player will follow in every possible attainable (erzielbare) situation in a game. Such moves may not be random, or drawn from a distribution, as in the case of mixed strategies.

Bei einer pure strategy gibt es keine Wahrscheinlichkeitsverteilung zwischen den actions bzw strategies. eine strategy wird entweder gewählt oder nicht. also p=1 oder 0 und nichts dazwischen. 

Two firms produce identical products (perfect substitutes) at constant unit costs c. 

They have to decide about the quantity they should choose. Payoff (mit nachfragefunktion eingesetzt) erste ableitung für q1 und q2. Das ergibt jeweils Best response function. 

q(m) < q(co) <q(pc)

Ein Spiel bei dem die Spieler durch Koordination die höchsten payofss bekommen. zB Battle of the sexes, Betamax vs VHS etc, auch stag hare... 

players are better of if they can coordinate....

 Also es gibt 2 NE und die payoffs in den kasteln wo kein NE ist, sind niedriger...

Wenn 2 nash equilibria vorhanden sind, in denen die Spieler die gleichen actions wählen wollen (im Nash Equlibrium). zB Stag Hare, Betamax, VHS 

If there is more than one nash-equlibrium, there may be certain NE's that are more likely to be choosen (because they are somehow special or more relevant to them --> kann auch wegen höheren payoffs für jeden spieler im diesem nash-gleichgewicht sein, aber auch wegen banalen sachen vgl kartenspiel), although the players cannot communicate. These "more likely choosed" NE's are called focal points. 

• competition over prices
• sehr zache competition, weil NE bei p=MC

beispiel: MC=o payoff: U=q*p-q*c 

Wenn Firma 1 einen Preis von 10 festsetzt, dann würde Firma2 zu einem Preis <10 produzieren und wegen perfektem substitut den gesamten profit alleine einstreichen. 

Wenn Firma 1 unter p<c produziert, dann würde Firma2 einen höheren Preis wählen. Das ganze pendelt sich also bei p=mc ein, denn dort will keine Firma einen anderen Preis verlangen... 

A mixed strategy is a probability distribution over all actions of the player. ( probability, that a certain action is choosen must be between 0 and 1 (zB 1>q>0) oder: "a strategy that assigns a prbability distribution to each of the other player's action."

A strategy in which the players choose their actions with a certain probability. This probability is between 1 and 0. 

zB bei penality kick α1(left)= 1/2 α1(right)= 1/2 oder kurz α1= (1/2,1/2) 

If p=1 its a pure strategy!!! 

"A strategy that assigns a probability to each of the player's actions" 

if there is one nash equilibrium in a game with pure strategies, it leads to a static and steady outcome --> all players choose the nash-equilibrium actions, because no player can gain from deviating from his optimal action a*i. Every play leads to the same outcome (=NE)

"players have preferences over probability distributions" --> Es gibt wahrscheinlichkeitsverteilungen die uns lieber und solche die uns weniger lieb sind. Welche uns am liebsten ist bestimmt sich nach den expected payoffs

"In a strategic game with vNM preferences, players have preferences over the expected values of a payoff function" 

Entscheidungen unter Unsicherheit werden mikroökonomisch oft als Lotterie modelliert. Der Nutzen der Wahl einer Alternative ist hier nicht unmittelbar bekannt. Statt einer Nutzenfunktion wird daher eine Erwartungsnutzenfunktion (auch VNM-Nutzenfunktion) für die Modellierung der Präferenzen des Akteurs eingesetzt.

Dabei wird der Erwartungswert über eine (typischerweise eindimensionale) Nutzenfunktion für die einzelnen Alternativen als Nutzenwert definiert. Die Nutzenfunktion der jeweiligen Alternativen und deren Wahrscheinlichkeitsverteilung bestimmen daher den Nutzen einer Lotterie: Erwartungsnutzen ist einfach der Erwartungswert des Nutzens der Alternativen. Eine solche Nutzenfunktion wird auch als Von-Neumann-Morgenstern-(Erwartungs)-Nutzenfunktion bezeichnet.

Achtung: bei mixed strategy stochastic, bei pure static! stochastik heißt sowas wie "von Wahrscheinlichkeit abhängig"

A stochastic steady state is a state in a game with mixed strategies, where no player  wants to change their strategy anymore --> mixed strategy nash equilibrium

• a strategy where no player wants to change his strategy given  the other players strategy in NE, because (wieder given strategy) the expected payoffs are at least as high as 

Payoffs are calculated by weighing the payoffs (by multiplikation) with the pprobability. Therefore they must have cardinal (not ordinal interpretation. players in ms are also risk neutral! 

• expected payoffs are at least as high as high as in any other mixed strategy, given that player i-1 (all other players) choose α*
  • Ui(α*)>=U(αi, αi-1)
  • "In a mixed strategy Nash equilibrium, each player's expected payoff to the mixed 
    strategy α∗i is at least as large as her expected payoff to any other mixed strategy αi, 
    given that all other players choose α∗!!." 
• -->therefore no player can gain by deviating from his strategy α*, given the other player chooses α* 
   
• A mixed strategy profile (zB α = ((1/2,1/2), (1/2,1/2))is a NE, if every player chooses a mixed strategy which is a best response to the other player's mixed strategy. ////////// every player's mixed strategy is a best response to the other players mixed strategy
   
• in NE, The expected payoffs to all actions chosen with positive probability must be equal (player is indifferent between actions. --> U1(a1, α2)=U1(a2, α2)  U1(Left, α2)=U1(Right, α2) 
  • "In the mixed strategy nash equilibrium, every player's expected payoff to each of his actions is the same. 

bei pure zB B1(quiet)=(fink) --> (a set of) action(s), which are the best (iSv payoffs are at least as high as in any other action) action (answer) to an action of the other players (a-1)

bei mixed: zB B1(p=1)= q=1 a mixed strategy which is the best "answer" 

A strategy profile is a combination of strategies of all players (one strategy per player). Each Nash equilibrium is for axample an strategy profil (except in pure strategies, where we talk about action profiles). 

e.g. a=((1/2,1,2), 1/2,1/2)) --> strategy profile in mixed strategy

s=(In, Agressive) or s=(Offer,Offer), Buy) 

a game in which the players move sequentially (after each other) (--> Im UNterscheid zu strategic game). And at least some players can observe the previous moves. IT doesn't necesseraly have to be that one person moves first. By making a credible strategic commitment you "produce" a sequential game.  An extensive game with perfect information can be represented in a game tree. An eg with perfect information consists of: 

• players 
• terminal histories 
• a player function (shows which player moves when ["assigns a player to every nontermal history"] zB P(in)=INcumbent
• preferences over terminal histories (immer am ende der history stehen preferences)

A history basically shows in sequential games what can happen when (after which action) in a game. So it shows the sequence (Abfolge) of actions. 

terminal: a sequence of actions that shows what may happen in the game from the start of the game to an action that ends the game

Nonterminal history: does not show the sequence of actions from the beginning until the end of a game (like terminal history); the game still continoues. So after a nonterminal history there are still actions or choices to be made by a player

empty history: basically an empty history says that nothing happened before. in sequential games it means that after an empty history the game starts

A player function shows which player moves after which nonterminal history in a sequential game. So it shows who moves when. 

e.g. P(In)=INcumbent

a strategy is a complete plan of actions. That means it shows a player what he shall do at every decision node/information set (after every nonterminal history), after /at which he has to move. Although the node is not reached! 

A strategy defines a set of moves or actions a player will follow in a given game. A strategy must be complete, defining an action in every contingency (Eventualität) , including those that may not be attainable in equilibrium. 

A strategy tells a player what to do for every possible situation throughout the game. So a strategy tells the player (immer einem!) what to do after each history that has occured if it is his move. 

Spieler als Roboter vorstellen, der immer wissen muss wo er hin muss. 

a combination of the strategies of all players zB s= (innovate, (copy,don't innovate). ---> matrix vorstellen. 

in sequential move games das was schlussendlich passiert (the result of the game), bei einem bestimmtem strategy profile. 

The terminal history that occurs at a certain strategy profile (combination of straegies of the players). 

Which terminal history occurs depends on the strategy profile (the combination of strategies of all players) 

• O(s*)>= O(si, s*-i) 
• ui (O(s*))>= ui (O(si, s*-i)) 

In the NE no player wants to change his strategy, given that the other players choose a strategy thats in the NE, because the strategy in the NE is a best response to the other players strategy (in NE). That means that no player can gain payoff by choosing another strategy, given the NE-strategy (s*) of the pther plaers 

• A strategy profile that induces a nash equilibrium in every subgame of the game. That means that no player can gain by deviating from his strategy s*, given that the other players choose s-i*. 
• s* is a best response to each available action the other players can take
• a strategy profile with the property, that in no subgame any player can do better than choosing this strategy, given that the other players choose "the subgame perfect strategy"

Jeder player wählt in jedem subgame seine optimale strategy, given die optimal gewählte strategie der anderen

The subgame perfekt equilibrium can be found by backward induction. 

a game tree is a graphically represents an extensive game (sequential game). It provides information about the players, payoffs, strategies, and the order of moves. The game tree consists of nodes (or vertices), which are points at which players can take actions, connected by edges, which represent the actions that may be taken at that node. An initial (or root) node represents the first decision to be made. Every set of edges from the first node through the tree eventually arrives at a terminal node, representing an end to the game. Each terminal node is labeled with the payoffs earned by each player if the game ends at that node.

a game within the game.

• a subgame starts at a singel node (after a nonterminal history)
• it comprises everything that follows after this node (the whole part of the game that remains
• it does not break up any information set (in case of imperfect information)

With backward induction it is possible to find the subgame perfekt equilibrium of sequential bzw extensive games. One has to start at the end of the game (last subgame(s)). There you start to determine the optimal action(s) of the player. Taken that action(s) as given determine the optimal action of the player who moves at the next (legth 2) subgame. In this process you find a NE in every subgame of the game, which leads to finding the subgame perfect equilibrium (NE which induces a NE in every subgame of the game)