Week 1
Game theory
Game theory = a tool for modeling multi-person decision situations/ interactions (that
involve joint and/ or conflicting interests)
- Interactions make game theory different from one-person decision theory
- Game theory does not offer any specific answers to any specific situation.
It says something like ‘These are the things to take into account’
Separating equilibrium = An equilibrium in which agents with different characteristics
choose different actions
- In an insurance market high-risk agents and low-risk agents will choose different
insurance contracts
Pooling equilibrium = An equilibrium in which agents with differing characteristics choose
the same action.
- In an insurance market high-risk and low-risk agents choose the same insurance
contract
Branches of game theory
- Non-cooperative game theory
- Cooperative game theory
- Evolutionary game theory
Games are modeled in terms of the
- Strategies available to the players (noncooperative game theory)
- Outcomes that could be attained by coalitions of players (cooperative game theory)
Non-cooperative game
Consists of 5 ingredients:
1. Players
- Who can be a player? Anything
- Two key attributes: preferences (i.e. ranking of possible choices) & beliefs (i.e.
person’s capacity to form beliefs as to what others will do)
- Number of players: monopoly = 1, oligopoly = 2-4, perfect comp. = many
- Who are the players? A player chooses one out of two or more possibilities
2. Actions/ strategies
Actions = list of all the possible actions a player can choose from in the entire game
Strategy = a specification of an action for each observable history of the game
or a specification of an action for each information set.
= a complete plan of action, function that specifies for each information set of a
player, what action he would take if the game reached the information set (A, B,C)
Game tree / Extensive form = a tree with branches representing the actions available
to particular players as a consequence of actions taken earlier in the tree
- information sets = sets of nodes which indicated what a player knew about
earlier decisions when it was his turn to move
, o all the nodes in an information set are indistinguishable to that player
at the time when the choice of action was demanded
- Node indicates that a player has to make a decision;
- Branch represents a choice/ action.
A continuous variable (e.g. a share of the profits, price of a product) is reflected by a
triangle. E.g. bargaining:
3. Payoffs
Payoffs in the extensive form are at bottom of tree
Strategic form = matrix
- The entries of the strategic form consist of all the strategies of the players
- First player goes on the left of the matrix, second player above
4. Information structure
= The knowledge that each player has about the game, such as the rules, the payoffs,
and the actions of others (i.e. sequential or simultaneous game)
5. Rules of the game
Who goes first
Nash equilibrium
At the core of game theory are:
- Maximization = individuals try to choose their best feasible option
- Equilibrium = individuals try to choose their best feasible option when interacting
with others
Approach for solving a game
- Equilibrium thinking: investigates for every player whether an action is the best
action for this player, while taking into account the choices of the other players.
- Determination of the endogenous (= dependent) variables (i.e. the equilibrium
choices of the players)
Equilibrium specifies the values of the endogenous variables as a function of the
exogenous (= independent) variables (aspect of the specification of the problem, i.e.
players, actions, payoffs, information structure, rules)
- Equilibrium analyses
o Comparative statics analysis/ hypothesis testing/ empiricism = determines
how a change in an exogenous variable changes the endogenous variables
o Efficiency analysis / prescriptive analysis = An outcome is efficient when it
creates the highest possible value for all players involved
- Behavioral assumptions: equilibrium (analyses) are performed, given certain
behavioral assumptions
Nash equilibrium = A strategy for each player which maximizes his payoff, given the strategy
of all other players
- Gives strategies for each player, not payoffs
- Equilibrium need not be efficient, all players have no incentive to deviate unilaterally
- NE is an interactive equilibrium
, Efficiency
An equilibrium outcome is efficient when the sum of the equilibrium payoffs of the players is
maximized. Rational decisions (i.e. NE) may result in an inefficient outcome / equilibrium in
strategic situations
Subgame perfect equilibrium
NE in strategic form: determine for each column and each row which of the strategies
results in the highest payoff.
Concept of NE in strategic form does not always point to a unique pair of strategies,
however some strategies are not credible (player might always get higher payoff with
different strategy) equilibrium has to exclude non-credible threats
Principle of backward induction: The idea that an equilibrium strategy should also make
sense in contingencies that do not arise during the actual play
Subgame perfect equilibrium = a NE where the strategies are NE in every subgame (use
extensive form & backward induction)
- Every SPE is a NE, but not every NE is a SPE (NE with non-credible threats do not
survive the SPE requirement of being a NE in every subgame)
Determining equilibrium
Calculate NE with the strategic form representation.
Calculate SPE with the extensive form representation
(5) Rules of the game, who goes first
Arrow’s impossibility result
- There is cycling in terms of the preference of the majority, i.e. no stable outcome
McKelvey-theorem
- Ex.: 3 judges decide about the involvement of a suspect in a crime and the penalty.
- 3 penalties: Death, Life imprisonment, Acquittal. Each judge has different order of
penalty preferences.
- 3 ‘rules of the game’: Status Quo system (1st: guilty? 2nd: penalty?), Mandatory
sentencing (1st: penalty? 2nd: guilty?), Roman tradition (1st: Death? 2nd: Life
imprisonment?)
o The sequence of decisions (choice of rules) may have a large impact on the
outcome, each rules gave different outcome
- Theorem: Every possible outcome of a democratic decision process can be
established by an appropriate choice of the layers in a decision procedure.
Relationship with Arrow’s impossibility theorem: Specifying rules of the game, by
delineating a sequence of decisions favors a certain outcome of the game
Game theory
Game theory = a tool for modeling multi-person decision situations/ interactions (that
involve joint and/ or conflicting interests)
- Interactions make game theory different from one-person decision theory
- Game theory does not offer any specific answers to any specific situation.
It says something like ‘These are the things to take into account’
Separating equilibrium = An equilibrium in which agents with different characteristics
choose different actions
- In an insurance market high-risk agents and low-risk agents will choose different
insurance contracts
Pooling equilibrium = An equilibrium in which agents with differing characteristics choose
the same action.
- In an insurance market high-risk and low-risk agents choose the same insurance
contract
Branches of game theory
- Non-cooperative game theory
- Cooperative game theory
- Evolutionary game theory
Games are modeled in terms of the
- Strategies available to the players (noncooperative game theory)
- Outcomes that could be attained by coalitions of players (cooperative game theory)
Non-cooperative game
Consists of 5 ingredients:
1. Players
- Who can be a player? Anything
- Two key attributes: preferences (i.e. ranking of possible choices) & beliefs (i.e.
person’s capacity to form beliefs as to what others will do)
- Number of players: monopoly = 1, oligopoly = 2-4, perfect comp. = many
- Who are the players? A player chooses one out of two or more possibilities
2. Actions/ strategies
Actions = list of all the possible actions a player can choose from in the entire game
Strategy = a specification of an action for each observable history of the game
or a specification of an action for each information set.
= a complete plan of action, function that specifies for each information set of a
player, what action he would take if the game reached the information set (A, B,C)
Game tree / Extensive form = a tree with branches representing the actions available
to particular players as a consequence of actions taken earlier in the tree
- information sets = sets of nodes which indicated what a player knew about
earlier decisions when it was his turn to move
, o all the nodes in an information set are indistinguishable to that player
at the time when the choice of action was demanded
- Node indicates that a player has to make a decision;
- Branch represents a choice/ action.
A continuous variable (e.g. a share of the profits, price of a product) is reflected by a
triangle. E.g. bargaining:
3. Payoffs
Payoffs in the extensive form are at bottom of tree
Strategic form = matrix
- The entries of the strategic form consist of all the strategies of the players
- First player goes on the left of the matrix, second player above
4. Information structure
= The knowledge that each player has about the game, such as the rules, the payoffs,
and the actions of others (i.e. sequential or simultaneous game)
5. Rules of the game
Who goes first
Nash equilibrium
At the core of game theory are:
- Maximization = individuals try to choose their best feasible option
- Equilibrium = individuals try to choose their best feasible option when interacting
with others
Approach for solving a game
- Equilibrium thinking: investigates for every player whether an action is the best
action for this player, while taking into account the choices of the other players.
- Determination of the endogenous (= dependent) variables (i.e. the equilibrium
choices of the players)
Equilibrium specifies the values of the endogenous variables as a function of the
exogenous (= independent) variables (aspect of the specification of the problem, i.e.
players, actions, payoffs, information structure, rules)
- Equilibrium analyses
o Comparative statics analysis/ hypothesis testing/ empiricism = determines
how a change in an exogenous variable changes the endogenous variables
o Efficiency analysis / prescriptive analysis = An outcome is efficient when it
creates the highest possible value for all players involved
- Behavioral assumptions: equilibrium (analyses) are performed, given certain
behavioral assumptions
Nash equilibrium = A strategy for each player which maximizes his payoff, given the strategy
of all other players
- Gives strategies for each player, not payoffs
- Equilibrium need not be efficient, all players have no incentive to deviate unilaterally
- NE is an interactive equilibrium
, Efficiency
An equilibrium outcome is efficient when the sum of the equilibrium payoffs of the players is
maximized. Rational decisions (i.e. NE) may result in an inefficient outcome / equilibrium in
strategic situations
Subgame perfect equilibrium
NE in strategic form: determine for each column and each row which of the strategies
results in the highest payoff.
Concept of NE in strategic form does not always point to a unique pair of strategies,
however some strategies are not credible (player might always get higher payoff with
different strategy) equilibrium has to exclude non-credible threats
Principle of backward induction: The idea that an equilibrium strategy should also make
sense in contingencies that do not arise during the actual play
Subgame perfect equilibrium = a NE where the strategies are NE in every subgame (use
extensive form & backward induction)
- Every SPE is a NE, but not every NE is a SPE (NE with non-credible threats do not
survive the SPE requirement of being a NE in every subgame)
Determining equilibrium
Calculate NE with the strategic form representation.
Calculate SPE with the extensive form representation
(5) Rules of the game, who goes first
Arrow’s impossibility result
- There is cycling in terms of the preference of the majority, i.e. no stable outcome
McKelvey-theorem
- Ex.: 3 judges decide about the involvement of a suspect in a crime and the penalty.
- 3 penalties: Death, Life imprisonment, Acquittal. Each judge has different order of
penalty preferences.
- 3 ‘rules of the game’: Status Quo system (1st: guilty? 2nd: penalty?), Mandatory
sentencing (1st: penalty? 2nd: guilty?), Roman tradition (1st: Death? 2nd: Life
imprisonment?)
o The sequence of decisions (choice of rules) may have a large impact on the
outcome, each rules gave different outcome
- Theorem: Every possible outcome of a democratic decision process can be
established by an appropriate choice of the layers in a decision procedure.
Relationship with Arrow’s impossibility theorem: Specifying rules of the game, by
delineating a sequence of decisions favors a certain outcome of the game
