Topics: Semi-Separating Equilibrium, Partially-Pooling Equilibrium

(semi-separating is the more common label)

It was a terrorist group as Robust with probability .4 and Vulnerable with probability .6 the group

then chooses whether to commit an attack or not if the group attacks

then player 2 who is the target chooses whether to resist or ignore payoffs are as follows

If player 1 does not attack everyone receives a status quo outcome of 0

If player 1 attacks and player 2 ignores then player 1 gets a point for a successful attack and player 2 loses a point for the same reason

The key difference between the types is what happens following resistance the vulnerable group will completely fall apart

#### Robust Type

Let’s get to solving this we begin by noting that the Robust type clearly must attack it receives either 3 or 1 by doing so in contrast it receives 0 by not attacking

Thus no matter how player 2 acts the Robust type finds attacking more profitable

#### Vulnerable Type

Now let’s see if we can describe the vulnerable types action using the types of equilibria

we have covered previously suppose the vulnerable type separates by not attacking

then conditional on observing an attack player 2 knows it is facing the Robust type it chooses to ignore and receive -1 instead resisting and receiving -3 given this does the vulnerable type have a profitable deviation indeed it does bluffing is just too attractive

* Bluffing: a strategical method of demonstrating one’s unpredictability

#### Separating?

If it were to attack player 2 would ignore under the False assumption that player 1 is Robust

This allows the Vulnerable type to receive one which is better than the 0 it earns by maintaining its strategy

#### Pooling?

**u(resist) = .4(-3) + .6(2) = 0u(ignore) = .4(-1) + .6(-1) = -1**

for a total payoff of 0 if she ignores she receives negative one regardless of the opposing type

Because 0 is greater than -1 she chooses to resist given this does the vulnerable type have a profitable deviation

It does once more now the vulnerable types Bluff backfires it earns negative two for sticking with its strategy whereas it can secure zero by not attacking as such we cannot find any equilibria given our current tools

### Semi-Separating Equilibrium

- Each type does not take the same action
- Each type does not take a distinct action

This is where semi-separating equilibrium comes into play it is best to define semi-separation in terms of what it is not a set of strategies where each type takes the same action

It is also not a set of strategies where each type takes a distinct action

rather sometimes a type will mimic another type and sometimes it won’t in our game if the vulnerable type is not playing a pure strategy

#### Remember me Indifference Conditions

indifference conditions when solving for a semi separating equilibrium

The vulnerable type mixes for a player to be willing to mix all strategies within the mixture must produce the same expected payoff

otherwise one option would be better and the player would always want to pursue that option instead

notice that attacking can give either the vulnerable type of payoff of negative two or one not attacking generates zero

thus to make the vulnerable type indifferent player two must mix between resist and ignore

and if we remember our indifference conditions, we can quickly solve for player two’s mixed strategy

**u(attack) = \(\sigma_R(-2) + (1-\sigma_R)(1) = 1 – 3\sigma_R\)u(no attack) = 0**

let Sigma R be the probability she resists

then we need to find the value of Sigma R such that the utility for attacking equals the utility for not attacking

attacking generates negative two Sigma R a portion of the time and 1, 1 – Sigma R portion of the time

not attacking gives the vulnerable type 0

**\(1-3\sigma_R = 0, \sigma_R = 1/3\)**

if we work out the math Sigma R equals one-third(1/3)

that’s what generates indifference

this will be player two’s equilibrium strategy

we still don’t know the vulnerable types strategy however

what we do know is that player two must mix and for player two to mix she also must be in difference between her two pure strategies

what makes this slightly complicated is that player two’s belief depends on the vulnerable types mix strategy

for now let’s call her belief P this represents her belief that she is facing a robust type

**u(resist) = p(-3) + (1-p)(2) = 2 – 5pu(ignore) = p(-1) + (1-p)(-1) = -1**

if she resists she earns – 3 P portion of the time

and to 1 – P portion of the time

if she ignores she receives -1 regardless

**2 – 5p = -1, p = 3/5**

after doing a little bit of math we see that she is indifferent when her belief that player 1 is robust exactly equals 3/5

as we alluded to a second ago the trick here is that her belief forms endogenously based on the vulnerable types strategy

This marks the first time we need to use Bayes rule in a non trivial way for a perfect Bayesian equilibrium

We are looking for the vulnerable types mixed strategy that creates a p-value exactly equal to 3/5

**\(\frac{3}{5} = \frac{(.4)(1)}{(.4)(1) + (.6)(\sigma_A)}\)**

by Bayes rule her belief that player 1 is robust equals the probability player 1 is robust and attacks divided by all of the possible ways she could observe an attack

well 40% of the time player 1 is robust and always attacks

this gives us the numerator

we also put that in the denominator but there is another way player 2 can observe an attack

60% of the time player 1 is vulnerable and it attacks Sigma A portion of the time

so we add that to the denominator

**3(.4)(1) + 3(.6)(\(\sigma_A\)) = 5(.4)(1)1.2 + 1.8\(\sigma_A\) = 21.8\(\sigma_A\) = .8\(\sigma_A\) = 4/9**

from here it is just a matter of solving for Sigma A

working through the math we arrive at Sigma a equal to 4/9

by exploiting the necessary indifference conditions we have arrived at all of the elements of a perfect bayesian equilibrium

### A Semi-Separating Equilibrium

- Robust type attacks
- Vulnerable type attacks with probability 4/9
- After observing Player 1 attack, Player 2 believes he is robust with probability 3/5
- Player 2 resists with probability 1/3

to recap the robust type always attacks the vulnerable type only attacks with probability 4/9

after observing player in one attack player two believes he is robust with probability 3/5

she then resists with probability 1/3

because of all the indifference conditions the algorithm to solve for semi-separating equilibria is tougher than pooling and separating equilibria

- Reference: William Spaniel, Game Theory 101 (#82): Semi-Separating Equilibrium/Partially-Pooling Equilibrium, May 20, 2019, https://youtu.be/h3JQYd4BFGk