UNIVERSITY COLLEGE LONDON
DEPARTMENT OF ECONOMICS
Economics BSc (Econ)
Second Year – Term 1
MICROECONOMICS
ECON0013
Rodrigo Antón García
rodrigo.garcia.20@ucl.ac.uk
London, 2021
,
, Contents
Topic 1 – Games in Strategic Form.
o Topic 1.1 – Describing Games in Strategic Form. 1
o Topic 1.2 – Dominance and Nash Equilibrium. 3
o Topic 1.3 – Finding Nash Equilibria. 7
Topic 2 – Games in Extensive Form.
o Topic 2.1 – Extensive Form Games. 14
o Topic 2.2 – Nash Equilibrium and Backwards Induction. 22
o Topic 2.3 – Subgame Perfect Equilibrium. 26
Topic 3 – Monopoly and Oligopoly.
o Topic 3.1 – Monopoly. 31
o Topic 3.2 – Static Oligopoly. 40
o Topic 3.3 – Dynamic Oligopoly. 51
Topic 4 – Perfect Competition.
o Topic 4.1 – Production Sets and Functions. 58
o Topic 4.2 – Profit Maximisation. 64
o Topic 4.3 – Cost Minimisation. 68
o Topic 4.4 – Examples. 72
Topic 5 – Adverse Selection.
o Topic 5.1 – Lemons and Risk. 76
o Topic 5.2 – Insurance and Market Failure. 79
o Topic 5.3 – Two-part Tariffs. 86
o Topic 5.4 – Optimal Screening. 89
,Topic 6 – Moral Hazard.
o Topic 6.1 – Introduction and Bank Loans. 95
o Topic 6.2 – Contracts and Risk Neutrality. 100
o Topic 6.3 – Contracts and Cost of Risk. 103
Topic 7 – Economic Design and Mechanisms.
o Topic 7.1 – Mechanisms. 107
o Topic 7.2 – Auctions. 116
,Microeconomics – ECON0013 Rodrigo Antón García
ECON0013: MICROECONOMICS SUMMARY – TERM 1
Topic 1 – Games in Strategic Form.
o Topic 1.1 – Describing Games in Strategic Form.
A) Games in Strategic Form: Description and Examples.
A game in strategic form consists of several elements. These include the following:
The First Element is the selected List of Players; ! = 1,2, … , '. ' can be finite or infinite.
Here the names of the players are just numbers but they can be any entity you are
interested in studying.
Examples of Lists:
- In Rock-Paper-Scissors there are two players. ! = 1,2.
- In an oligopoly with ) firms there are ) players. ! = 1,2, … , ).
- In bargaining there is a buyer and a seller, two payers. ! = *+,-., /-00-..
The Second Element is a set of Actions or Pure Strategies; a description of all the actions
each player ! can take, this is written as 1! . We write the typical action as /! 3 1! .
Examples of Action Sets:
- In Rock-Paper-Scissors, a player has three pure strategies: so
/! 3 {5678, 9:;-., 17!//6./} = 1! , for ! = 1,2.
- In an oligopoly, each firm, !, will choose a quantity, 0 ≤ ?! the set 1! is the set of
all positive quantities.
- In bargaining, the buyer proposes a price 0 ≤ ; ≤ 1 and the seller makes a
proposal too. 1"#$%& = {0 ≤ ; ≤ 1}, 1'%((%& = {0 ≤ ; ≤ 1}.
The actions are summarized in an Action or Strategy Profile. This is a list for all actions
of players in the game. / = (/) , … , /* ) 3 1.
Examples of Strategy Profiles:
- In Rock-Paper-Scissors, an example of a profile is / = (17!//6./ , 9:;-.) –
player 1 does S and 2 does P.
- In an oligopoly, a profile is a list of quantities for every firm. Each firm, !, will
choose a quantity, (?) , ?+ , . . . , ?, ).
- In bargaining, a profile is two prices (;- , ;. ).
The Third Element of a game in strategic form are the Payoffs; the utility or profit the
players get from their actions or pure strategies. We can write this as a utility/payoff
function +! (/) = +! (/) , … , /! ).
This is a function that tells you what each player’s payoff is for each possible action
profile. There are usually determined by the rules of the game.
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,Microeconomics – ECON0013 Rodrigo Antón García
Examples of Payoffs:
- In Rock-Paper-Scissors: two players ! = 1,2, each with three actions 1! = {5, 9, 1}.
- In oligopoly: firms ! = 1,2, … , ) , each choosing an output 0 ≤ ?! , and
9.6B!C/ 6B B!.D ! = 5-E-)+- − G6/C/.
1) Suppose that each firm’s costs only depend on their output G6/C/ = 7(?! ).
2) 5-E-)+- = 9.!7- × I+:)C!C, = 9(?) + ?+ + ⋯ + ?, ) × ?! . Price depends on
the total output ?) + ?+ + ⋯ + ?, .
3) 9.6B!C/ 6B L!.D ! = ?! 9(?) + ?+ + ⋯ + ?, ) − 7(?! ).
To make this more concrete suppose that 9 = 50 − (?) + ?+ + ⋯ + ?, ), I = ?! , and that
7(?! ) = 2?! . Then, 9:,6BB/9.6B!C = ?! (50 − (?) + ?+ + ⋯ + ?, )) − 2?! .
Is a perfectly competitive market a game?
No – in a perfectly competitive market agents choose how much to demand or supply.
They choose quantities. There is no way to explain the determination of price,
participants are price-takers, they just choose quantities. To turn a perfectly competitive
market into a game we would need a player out there that chooses the price.
Random Actions and Mixed Strategies.
We want to allow the players to randomize in their choice of an action, such as in Rock-
Paper-Scissors. We call these random actions Mixed Strategies.
A Mixed Strategy exists in a strategic game, when the player does not choose one
definite action, but rather, chooses according to a probability distribution over his actions.
It is not necessarily the case that players actually behave randomly. What is important is
what player A thinks player B is going to do. As player A may not know what B is going
to do, B’s actions should be treated as being random by player A.
The set of all the player’s mixed strategies is the set of all probability distributions on
their pure actions. We write the mixed strategy of player ! as O! . The profile of mixed
strategies for all players is written as O = (O) , … , O* ).
Examples:
- In Rock-Paper-Scissors, 1! = {5, 9, 1} :)P O! = {;, ?, 1 − ; − ?}.
- In oligopoly, or any a random quantity choice where a firm is choosing quantities
(random and positive), 1! = [0, ∞) a set of possible quantities mixed strategy is a
cumulative distribution function: L(S): Pr(?+:)C!C, ≤ S).
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,Microeconomics – ECON0013 Rodrigo Antón García
Payoffs from Mixed Actions.
A player’s payoff is an expectation taken over their random action and all the other
random actions of all their components. This expectation is taken assuming the players
randomise independently.
Example of Payoff from Mixed Actions:
- Suppose we think the row player is choosing Rock with probability ;, Paper with
probability ? and Scissors with probability 1 − ; − ?.
- If you choose Rock, you valuate this option by taking the expected value:
WS;-7C-P(5678) = 0 × ; + (−1) × ? + 1 × (1 − ; − ?) = 1 − ; − 2?
- Equally, if you choose Paper:
WS;-7C-P(9:;-.) = 1 × ; + 0 × ? + (−1) × (1 − ; − ?) = 2; + ? − 1
- And so, if you choose Scissors:
WS;-7C-P(17!//6./) = (−1) × ; + 1 × ? + 0 × (1 − ; − ?) = ? − ;
o Topic 1.2 – Dominance and Nash Equilibrium.
B) Dominance Arguments: Iterated Strict and Weak Dominance.
A new piece of notation is //! describes a list of actions for all the players who are not
player !. That is, //) = (/+ , /0 , … , /* ), //+ = (/) , /0 , … , /* ), etc.
- Strict Dominance.
A mixed strategy O! strictly dominates the pure action /!1 (weaker strategy) for player !, if
and only if, player !’s payoff when he plays O! and all the other players play the actions
/!1 is strictly higher than his payoff from /!1 against //! for any actions //! the others may
play.
+! (O! , //! ) > +! (/!1 , //! ), ∀ //!
Example:
- Suppose we consider two strategies, we only put in the row player’s payoffs as
that is all that matters for now.
) )
- O = Z+ , + , 0[ on Top, Middle, (and Bottom).
- / = (0,0,1) That being only bottom.
We can write the expected payoffs to these strategies the following way. Firstly, when
0 0
playing O, if opponent goes L, then we get +; and, if my opponent goes R, then we get +.
Similarly, when playing /, if we play Bottom, we get 1 if my opponent goes either L or R.
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, Microeconomics – ECON0013 Rodrigo Antón García
) )
The O = Z , , 0[ (Top-Middle) strategy is strictly better than what you get from going / =
+ +
(0,0,1) (Bottom), whatever the column player does.
Therefore, / is a strictly dominated strategy that would never be played.
Eliminating Strictly Dominated Actions.
Eliminating strictly dominated actions can allow us to make
strong predictions about what actions the players will use.
Here we have L strictly dominating R, so we can eliminate
this column, this is because 10 > 9 and 6 > 5.
Since R is never played, U strictly dominates D for the row
player given that 8 > 7. And so, we predict (`, a) for the
outcome of this game.
However, the row player may be very worried about the −100 because how do they
know the column player is rational?
Finding the Range of Mixed Strategies that Strictly Dominate an Action.
We will look at the following game:
? 1−? ⋯ 0 ⋯ ?,
b c ⋯ d ⋯ e2
; f ("!"# , $!$% ) ("!"& , &!$% ) ⋯ ("!"' , (!$% ) ⋯ ("!"(! , )) !$% )
1−; g (*!"# , $!$* ) (*!"& , &!$* ) ⋯ (*!"' , (!$* ) ⋯ ("!"(! , )) !$* )
⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
0 h ( !"# $!$+ )
+ , ( !"& &!$+ )
+ , ⋯ ( !"' (!$+ )
+ , ⋯ ( !"(! )) !$+ )
+ ,
⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
;, e3 ( & !"# , $!$(" )
) ( & !"& &!$(" )
) , ⋯ ( & !"' (!$(" )
) , ⋯ )
( & !"( , )) !$( )
! "
- ` and i, and, a and 5 represent strategies of the Row and Column players to
which we assign probabilities ;, 1 − ; and ?, 1 − ? respectively.
- ' and j represent the strategies of the Row and Column players that are going to
be dominated in this example.
- k4 and k5 represent the infinite number of strategies of the Row and Column
Player that are not dominated.
- `675 represents `’s payoff when Column plays a.
- a674 represents a’s payoff when Row plays 5.
We will assign probabilities ; and 1 − ; to ` and a, and probabilities ? and 1 − ? to a
and 5 respectively (If you had more than two strategies for the row player, you would
assign ;, probabilities ;) , ;+ and 1 − ;) − ;+ ).
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