In this lecture, we are going to introduce the main concepts about game theory beyond payoff matrix.
In fact, to have the maximum advantage 
by game theory, we need to investigate
3 main concepts, which are dominated options, 
Pareto efficient solution and Nash equilibrium point.
Let’s start with dominated options.
An option is said to be dominant when it offers
the player who makes this choice a greater pay-off
than if he choses any other option,
whatever his opponent decides.
Considering our example, we could notice 
that for Player 1, DVD is giving a payoff of 3
rather than 1 if Player 2 is selecting DVD;
moreover, DVD is giving a payoff of 4
rather than 2 if Player 2 is selecting blu-ray.
Thereby, DVD is a dominant option for Player 1.
On the contrary, we could not find a dominant option 
of Player 2: if Player 1 is selecting DVD,
Player 2 is preferring DVD (payoff of 3 rather than 2), 
but if Player 1 is selecting Blu-Ray,
Player 2 is preferring this standard too 
(payoff of 4 rather than 1).
As a matter of fact, is very useful to identify
dominant option, because in this situation
is possible to take rational decisions 
reducing uncertainty at all.
Second concept is Pareto efficient solution.
A solution is Pareto efficient if there is no other solution that can produce a better result for both players
and neither can improve
without the other’s pay-off getting worse.
It is a form of solution that considers 
the overall system and not the single player perspective.
Considering our example, 
we actually have 3 different efficient solutions.
(3;3) is Pareto efficient because any other
solution is improving performance for both the actors.
Moving for example to (2;4), payoff is improving
for Player 1 but decreasing for Player 2.
The same is true for solution (4;2).
In case of (1;1) actually payoff is decreasing
for both the actors.
Actually (2;4) and (4;2) are efficient solutions
as well for the same reasons.
Pareto efficient is a key concept to demonstrate
that sometimes simply following
the personal optimal solution is not the best way 
to achieve the system optimum.
This concept is fundamental specially to modelling
collaborations and partnerships through game theory.
Finally, the last concept is Nash equilibrium point.
In this point, no player taken on his own
has any interest in changing his/her decision,
taking the other party’s decision for granted.
Nash equilibrium point is also the game solution.
Considering our example, for Player 1 if
Player 2 is selecting DVD is better DVD (3
rather than 1); if Player 2 is selecting blu-ray
is better DVD (4 rather than 2).
On the other hand, for Player 2, if Player 1 is selecting DVD, is better DVD
(3 rather than 2); if Player 1 is selecting blu-ray
is better blu-ray (4 rather than 1).
As a matter of fact, we could notice 
that the equilibrium point is DVD for both the actors.
As a matter of fact, we know that DVD 
is a dominant option for Player 1 and so we could take
for granted that Player 1 is going to select that option: 
in case of a dominant option,
the equilibrium point is the best option 
of the other player.
In this case, the equilibrium point is by chance also 
an efficient solution.
This is a positive point because individual
optimal solution and system optimal solution
coincides, but it is not taken for granted.
