Prenote;
The purpose of this document is to attempt to embody the key aspects of each chapter studied,
highlight certain parts that the book or professor highlighted and to keep it as concise as
possible. Hence, we will not go into many examples, just the theory. Examples can be found in
the book, slides/lectures, or online. I will attempt to keep each summary within 2-3 pages, but if
a chapter is too large, this may not be possible. Feel free to use the table of contents to the left
to scroll through quicker.
Week One:
Pareto efficiency is the state in which you can not make anyone better off without making
anyone worse off. Hence, if there is waste it cannot be efficient, as it’s easy to make someone
better off without making someone worse off.
Game Theory:
Game theory is the formal study of decision-making where economic actors make choices that
can potentially affect the outcomes of the other actors. In a game you have strategic interaction;
Players They maximise their own payoff (self interest)
Outcomes or payoffs What a player would get/give up if they make a specific choice
over another.
Strategies Plans of actions
Rules Every game has rules!
We assume that people have common knowledge of rationality → everyone knows my payoffs
and the actions I take and everyone knows that everyone knows!
Payoff matrix
This matrix shows the payoffs for James and
Tom, the person on the side of the table is
always on the left, while the person on top
always displays his payoffs on the right.
Here the nash equilibrium is 1,1 or D,R
Nash equilibrium and strategy:
,Nash equilibrium is where both people are responding to the other in the best possible way. A
Nash strategy is given all strategies, the player maximises his/her utility with this response.
Dominant strategy and equilibrium:
A dominant strategy is the strategy that a player will follow no matter what. This strategy is
chosen because it will always yield the best outcome for that person. You get the equilibrium
when they follow a dominant strategy.
It’s possible that one person has a dominant strategy and the other doesn’t, it’d look like this:
The Prisoner’s Dilemma:
Standard Case:
Here we see a case of two prisoners. The Nash equilibrium
would be if both confess. This is because if one doesn’t and
the other does, the one not confessing would be in a very
bad place.
Why don’t they just not confess, and both do 1 year? Well
trust is an issue, one person could just backstab the other!
, Example of it not being a prisoner's dilemma.
Here we see that the Nash equilibrium would be top left. So
5,5. Why? Well for Tom it is always best to go up and for
James it is always best to go left! This is efficient and thus it
is NOT a prisoner’s dilemma. A prisoner’s dilemma leads to
an INEFFICIENT outcome and BOTH have a dominant
strategy!
Two Nash equilibria:
Here we can see that Sally and Harry would
rather be together than apart. So both going
to the theatre and both going to the soccer
game are both Nash equilibria. Again, we see
an efficient outcome!
Empirical evidence shows that if a penalty is higher, people will tend to almost always go for the
nash equilibria. The standard ‘better safe than sorry’ approach!
Repeated Games:
In any prisoner's dilemma game, if it is played multiple times, there will always be the same
Nash equilibrium, Confess and Confess!
However, if person A would play a tit for tat (example on the slides) then confessing in each
game is not a dominant strategy, as you can abuse this and get better off yourself! The example
in the slides shows how a tit-for-tat can lead to a better outcome. This can not be an equilibrium
though, as now James will not be best responding, so a tit-for-tat does not bring an equilibrium!
For him it would just be best to confess-confess.
Infinitely Repeated Games:
In most real life situations, this is the case… as you don’t know how many times you will play the
game. In this situation we have a grim-trigger-strategy. Which basically means ‘I will cooperate
as long as they did it in the previous round, if they once defect, then I will never cooperate
again.
, So let’s say James plays the grim-trigger. As
long as Tom cooperates, they will be in a
Nash equilibrium. If Tom doesn't do it once,
he will get 3 once and then 1 onwards, so he
does not want this. Short-term gains are
outweighed by long-term costs.
However, both D,D and C,C can be Nash equilibrium, grim-trigger just incentivises C,C.
Until now we have assumed that money keeps its value, however we all know it doesn't. Hence
δ ∈ (0, 1)
. If it is close to 1, the player is very patient, if it isn’t, the player is impatient. Hence, if we play
the grim-trigger, the future values are discounted, giving more value to the present values. The
𝑝𝑎𝑦𝑜𝑓𝑓
way we can find the present value is by doing: 1−δ
. If we were to defect in the first run (with
the example above) we would get 3 + 1 + 1 + 1 + 1… which would lead to an expected payout
𝑝𝑎𝑦𝑜𝑓𝑓2
of (𝑝𝑎𝑦𝑜𝑓𝑓1 − 𝑝𝑎𝑦𝑜𝑓𝑓2) + 1−δ
. If you want to see the maths behind this, the lecture slides
show it (it’s a simple geometric sequence, just the first number is different so we need to take
away 1. Hence, he will cooperate as long as;
2 1
1−δ
>2+ 1−δ
In this example, if it is larger than ½, he will cooperate.
Complete Plan of Action:
It describes which action they will take at each possible point in the game. In the example
below, Tom will have two strategies and James has four, dependent on tom, so tom must
specify these beforehand.
Here we see a tree-like representation!
