Definitions lectures 1-11 Flashcards
Strategic interdependence
action of one individual or group impact on others, individuals involved are aware of this
Formal study of decision making
several players must make choices that potentially affect interests of other players, decision includes the though of how it impacts others
Decision theory vs. Game theory
EX1
if everyone pays for their own bill= decision problem
If everyone agrees to split the bill= game problem
EX2
absolute grading system: decision problem to study for self
grading on a curve= this is a game, decision to study impacts others
Best choice/decision to make
highest payoff
case studies
analyzing situations and learning a recipe of how to play (many different situations, may be able to recognize parallelism)
game theory
focuses on general principles to explain why certain outcomes emerge (in a new situation you will recognize which principles to apply)
What is a game?
situation where each players actions affect others
Games that do not have strategic interaction
- games of pure chance: lotteries, slot machines
- games of pure skill: 100m sprint
- games without strategic interaction bw players: solitaire
Defining the game
- strategic environment
- the rules
- the assumptions
Strategic environment (I)
- players= everyone who has an effect on outcomes
- strategies= actions available to each player + define plan of action for every contingency
- payoffs= numbers (wins, losses) associated w each outcome + completely reflect interests of other players
The Rules (II)
- Timing of Moves
- Nature of conflict and interaction
- Informational conditions
- enforceability of agreements
Timing of moves
(Part of ‘The rules’ in ‘Defining a game’)
Are moves simultaneous or sequential?
SIMULTANEOUS MOVES
players have to figure out what the other is doing right now + you do not need to worry about retaliation
- analyzed w strategic forms of the game (payoff matrix)
- retaliation is important to consider: how ppl will react to your action
sequential moves
opponent (and/or yourself) reacting to other players (generally known) moves + have to account for future consequences
- analyzed w extensive forms of game trees
Do strategic games involve simultaneous moves or strategic?
Many require both!
Nature of Conflict
(“rules of game” in “defining a game”
- interest in conflict? zero sum or constant sum + coordination game
- will player interact once of many times? one shot games + repeated games
Zero sum (or constant sum)
Games where one player’s winning are the others losses
coordination games
games w multiple NE
- players have some common interests
- players act independently
- achieve a jointly preferred outcme
one shot games
play of the game occurs once– players don’t need to worry about retaliation + also can’t build up reputation or trust
repeated games
play of the game is repeated w same players + can build up reputations for toughness + can coordinate
*game can be zero-sum in short run but mutually beneficial in the long run
Informational conditions
(“rules of game” in ‘defining a game’)
- are some players better informed?
perfect information
if they know what has happened every time decision is made – no external uncertainty or strategic uncertainty
imperfect information
when one player knows more than the other
Enforceability of agreements
(‘rules of game’ in ‘defining a game’)
- can contracts be enforced (cooperative games)
cooperate games
games in which agreements are enforceable
non-cooperative games
games in which agreements are non-enforceable
** to understand how to reach cooperative outcome, first understand non-cooperative
The Assumptions (III)
- rationality
- common knowledge
Rationality (subset of assumptions)
- players understand aim to maximize own payoffs
- players flawless in calculating which actions will max. own payoff
common knowledge (subset of assumptions)
- each player knows rules of games
- knows each player knows the rules
- continues
Equilibrium concept
each player is using the strategy that is best response to strategies of other players
- likely outcome of game when rational strategic agents interact
equilibrium does not mean…
- that things don’t change over time (parameters of model can change)
- that everything is for the best (doesn’t mean every outcome of game is best)
payoff matrix (normal-form game)
table where strategies of one player are listen in rows and those of other players in columns
- helps determine dominant strategy and nash equilibrium if they exist
row
horizontal
column
vertical
Nash equilibrium
a pair of strategy choices (one for each player) that are ‘best responses to each others play’
- best response given highest payoff of what opponent has chosen
- no incentive to deviate
Ways to find nash equil.
- cell by cell inspection
2. Dominant strate
Not nash equilibrium if…
one of players would like to deviate
incentive to deviate
only if you will get something higher for a different choice
Golden Balls Game
more than one nash equilibrium
interpretations of Nash equilibrium
- no regret= no player regrets choice after observing other player
- self-enforcing agreement= both agree on NE strategy profile
- viable recommendations=
- transparency of reasons= if players rational, can duplicate strategy
Dominant strategy
best strategy for a player no matter what opposing players do
- always play dominant strategy bc expect opponent will use their dominant
strongly dominant strategy
no matter what rival does it always does strictly better than other available strategies
*only one strongly dominant strategy
weakly dominant strategy
no matter what rival does, it does equally well and sometimes strictly better than other available strategies
- can have more than one weakly dominant strategy
- never eliminate bc u can lose a NE
dominant strategy equilibrium
if both players have dominant strategy, then we have a dominant strategy equilibrium
- strongly DS: combination of only SDS
- weakly DS: combination of dominant strategies some weak some not
Symmetric games
if you have something happen to one player, will happen to other player
- if P1 has weakly dominant strategy one way, P2 will have it the same way
Strongly or weakly dominated
no dominant strategy equilibrium= remove outcome that gives lowest payoff– reduces size of game
iterated dominance equilibrium/elimination
an equil. found by removing strongly or weakly dominated strategies until one pair of strategies remains– game is dominance solvable
- careful when deleting weakly dominated strategies
Best response analysis
- nash equilibrium and best response means that it yields the player the highest payoff given what opponent has chosen
- finds all possible NE of a game
Nash equilibrium strategy
if game is repeated and played by ‘populations’ observations of experiences over time make new players find the nash equilibrium
Beliefs (w complete information)
each players does not know actual choices of others but has beliefs about other players’ actions, as games played beliefs are updated
complete information
players know payoffs of other players
best response dynamics
both players play best response strategies to each other and believe the opponent will play that strategy
Nash equilibrium: set of strategies
- each player has correct beliefs about strategies of other
- strategy of each player is best response given beliefs about strategies of others
pure coordination games
some games have more than one equilibrium
No nash equilibrium in pure strategy
sometimes best response analysis finds not equil. = no equil in pure strategy, but could have equil. in mixed strategy
EX/ tennis match as simultaneous move
pareto efficiency
an outcome if it is not possible to improve the payoff of one player without lowering payoff of the other
- one solution is the most efficient way
Pareto domination/ pareto superior
an outcome pareto dominates or is pareto superior to another if payoffs of one of more players are higher and none is lower
Types of simultaneous games
- pure coordination game
- assurance game
- battle of the sexes
- chicken games
pure coordination game
- payoffs of NE are better than other payoffs
- payoffs of each player are same in all NE
- *coordination is needed but only to ensure avoidance of non-equil.
Assurance game
both players have higher preference/ higher payoffs for one option= focal point
-coordination is needed to avoid non-equil + to achieve jointly preferred outcome
constant sum game
one where sum of payoff for any combination of strategies is equal to constant
– conflict of interest among players (higher payoff for one player means lower for other)
zero sum game (type of constant sum game)
if sum of payoffs for any combo of strategies is = to zero
convergence of expectations
situation where players in non-cooperative game can develop common understanding of strategies they expect will be chosen
focal point equilibrium
configuration of strategies for players in a game, emerges as outcome bc of convergence of players expectations on it
- equilibrium where players expectations converge
Battle of the sexes games
- players prefer different equilibrium= conflict of preferences
- highest risk of coordination failure
chicken games
- each player has one strategy that is ‘tough’ and one ‘weak’
- two-pure strategy NE: one weak one strong
- each player prefers equil. where other player is weak
- most liekly outcome= non-equil.
- conflict of interest
Prisoners Dilemma Game
- each player has two strategies: to cooperate or to defect from cooperation
- each player has a dominant strategy
- dominant strategy equil is worse for both players than non-equil.
Prisoners Dilemma Game
- each player has two strategies: to cooperate or to defect from cooperation
- each player has a dominant strategy
- dominant strategy equil is worse for both players than non-equil.
externalities: tragedy of the commons
overexploitation of free resources
- when individuals neglect the wellbeing of society in the pursuit of personal gain
Fiscal Battles
multinational decides where to locate based on taxation: want to locate where taxation is lower
SEQUENTIAL MOVES
ex/ matchsticks
- players take turns and perfect info implies that players known everything
- strategic thinking
backward induction
look forward and reason back
representing sequential games
who, what, how much
- use game theory or extensive form
game tree/extensive form
representation of a game in form of nodes, branches, w associated payoffs
node
point from which branches emerge, or where branch terminates in a decision tree
- initial node or root of game
- action node
- terminal node
branch
each branch emerging from node in a game tree represent one action that can be taken at that node
Strategies vs. moves
- move= single action taken by player at a node
- strategy= complete contingent plan of moves– defines actions at all posible nodes where players could play
subgame
smaller game embedded in complete game
- game that develops from all nodes after node 1
- full game is a subgame of game
Path of play and outcome
how the game goes and what the payoffs of players are
equilibrium found from backward induction
subgame perfect nash equilibrium
subgame perfect nash equil
is combo of strategies that yield NE in every subgame
- *a sequential game can have more than one SPNE
- actions that are not sequentially rational are eliminated
zermelos theorem
in finite, two-player ‘win-lose-draw’ games of perfect info. players can either
- one player has strategy that gurantees a win or both players have a strategy that guarantees draw
(ex/ tic-tac-toe or chess)
subgame emphasizes what?
that in every subgame, even those that are not being played, players still have to act rationally
All SPNE are NE, BUT…
not all NE survive backward induction
First mover advantage
first mover is leader
- player who is leader is not worse off
*leader had ability to commit to advantageous position than P2 has to adapt to it
(ex/ of two firms investing)
second-mover advantage
second mover has flexibility to dapt to others choices to achieve better outcome (Ex/ when firms are choosing prices and tennis game ex)
*reacting optimally
both players have advantage
both can do better under one set of rules of play than under another (ex/ IR and ECB–if IR moves first this will result in better outcomes for both players than those of others*whats more beneficial to IR is more beneficial for ECB)
both players have advantage
both can do better under one set of rules of play than under another (ex/ IR and ECB–if IR moves first this will result in better outcomes for both players than those of others*whats more beneficial to IR is more beneficial for ECB)
when will dominant-strategy equilibrium hold in both sim and seq games?
when both players have a dominant strategy
no change in outcome advantage
some games have same outcome in both types regardless of order of moves
- normally when both players have a dominant strategy, equil doesnt change– prisoners dilemma games
TWO STAGE GAMES
w both simultaneous and sequential games (uses matrices and game trees)
*use backwards induction to solve for first stage
pure strategies/ mixed strategy equilibrium
each player needs to keep other guessing, act randomly or unsystematically, uncertain
mixed strategy
random mixture among pure strategies
- assigns to each action a likelihood/probability of being selected
- induces lottery over possible outcomes of games
expected payoff
corresponding probabiity-weighted averages of payoffs from constituent pure strategies
NE in mixed strategies
list of mixed strategies, one for each player, such that choice of each is best choice=highest expected payoff, given mixed strategies of other players
mixed strategy NE in zero-sum or constant-sum games
- mixing is better than playing pure strategy
constant sum games
sum of players’ payoffs is constant, same for all their strategy combinations
opponents indifferent property
players needs to find a mixed strategy that other player cannot exploit, so the other player is indifferent bw moves
which pure strategy should be played?
opponent indifferent between her own strategies
if indifferent bw two possibilities
you can randomize
best mixed strategy
one that makes opponent indifferent between his own pure strategies
In Harry Sally game (game of sexes), which equilibrium is the best and why?
pure strategies equil is higher than mixed strategy BC mixed strategy gives confusion,randomizes on where you are going to go and both players want to end up in same place
chicken game mix or pure?
there is an advantage for players to mix, pure strategies isnt best response for sure
Beliefs and Responses
each player forms beleifs about probabiity of mixture that other is choosing and chooses best response to those
NE occurs in mixed strategies when…
the beliefs are correct
difference bw being uncertain and having incorrect beliefs
- incorrect beliefs can be updates
