Discussion between johnny09 and Herr K. - 2019-02-07

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3:09 AM

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Q: Mixed Strategies in Bayesian Games

I am confused about the following statement from Wikipedia: "A mixed strategy is an assignment of a probability to each pure strategy. This allows for a player to randomly select a pure strategy. Since probabilities are continuous, there are infinitely many mixed strategies available to a player...

game-theory bayesian-game

The quantifiers at the end of your criterion for equilibrium in Bayesian games seem wrong. It should have been for all $s_i'\in S_i$ and for all $i$.

@HerrK.You're right. I'll edit the question.

Also, in the second-to-last paragraph, did you mean $s_i:\Theta_i\to\Pi(A_i)$, i.e. with lower case $s_i$ as opposed to upper case? It matters.

@HerrK. The reference I am using uses an upper case for the function of mixed strategies. For example, $S$ is a mixed strategy and the strategy of a player $i$, $s_i(\theta_i)\in S_i(\Theta_i)$ for all $\theta_i\in\Theta_i$ is a probability distribution.

I think you're confusing two notational conventions and this led to your confusion about the interpretation of the symbol $S_i$. If you take $S_i$ to be the set of pure strategies for player $i$, then it'd make sense to write $s_i\in S_i$ (i.e. the pure strategy $s_i$ in the set of pure strategies $S_i$). If you take $S_i$ to be a state-dependent mixed strategy, then it is a function of the states; however it makes no sense to write $s_i\in S_i$ under this interpretation.

Btw, $s_{-i}\ne s_i$ should be removed from the definition of equilibrium in Bayesian games. It makes no sense. $s_{-i}$ is a vector of $n-1$ elements whereas $s_i$ is a scalar.

3:09 AM

@HerrK.You're right on the second comment. I edited my question. I don't think I'm confused as I am just following this paper's formulation "Transition Models of Equilibrium Assessment in Bayesian Game," 2015 by Kiminao Kogiso and in particular Definition 2, Equation (3).

You should have included the information of the paper in your original question. On page 5997, the first paragraph of section II.A, the paper says "$S$ is a mixed strategy." That is simply wrong. It is clear from the context that $S$ is the set of mixed strategy profiles, not any particular mixed strategy. $S_i(\Theta_i)$ is the set of state-dependent mixed strategies for player $i$.

I hope you don't mind that I moved the discussion here.

If $S_i(\Theta_i)$ is the state-dependent mixed strategies for player $i$ then what is $S_i$ in equation $(3)$? It has to be abuse of notation here, right?

Since I want to test the Bayesian Nash equilibrium inequality, I have been trying to explicitly write down the set $S_i$.

So far, I have $S_i=\{(\underline{a},\underline{\theta}),(\bar{a},\underline{\theta}),(\underli‌ne{a},\bar{\theta}),(\bar{a},\bar{\theta}),(\underline{a},\underline{\theta}),(\b‌ar{a},\underline{\theta}),(\underline{a},\bar{\theta}),(\bar{a},\bar{\theta})\}$.

Sorry, I made a typo. $S_i=\{(\underline{a},\underline{\theta}),(\bar{a},\underline{\theta}),
(\underline{a},\bar{\theta}),
(\bar{a},\bar{\theta}),
(\underline{a},\underline{\theta}),
(\bar{a},\underline{\theta}),
(\underline{a},\bar{\theta}),
(\bar{a},\bar{\theta})\}$.

Each parenthesis is a conditional probability

so, when we write $s_i(\theta_i)$ or $s_i'(\theta_i)$ we mean one of the elements in $S_i$, right?

and since these are mixed strategies, each element of $S_i$ is assigned a probability, a number between zero and one.

1 hour later…

4:38 AM

@johnny09: It's impossible to enumerate the entire set $S_i$ since there are uncountably many elements in it. It is a set of *mixed* strategies, not *pure* strategies. As the paper notes, each element $s_i(\theta_i)\in S_i(\Theta_i)$ is a vector of probabilities: $$s_i(\theta_i):=\begin{bmatrix}s_i(\underline{a}|\theta_i)\\s_i(\overline{a}|\theta_i)\end{bmatrix},\text{ where $s_i(a_i|\theta_i)\ge0$ and $s_i(\underline{a}|\theta_i)+s_i(\overline{a}|\theta_i)=1.$}$$ A few elements in the set are $$s_i(\overline\theta)=\begin{bmatrix}0.1\\0.9\end{bmatrix},\quad s_i(\overline\theta)=\begin{bmat…

Seems like the chatroom does not have MathJax rendering. I'll post a screenshot of the above codes

user image

It is, however, possible to list all the pure strategies in $S_i$; they are the four degenerate distributions over the two actions

user image

as a sidenote, you may want to use this to get mathjax rendering math.ucla.edu/~robjohn/math/mathjax.html

so, this statement from wikipedia: Since probabilities are continuous, there are infinitely many mixed strategies available to a player.

is correct

i am confused now because how do we compute the Bayesian nash equlibrium? we have to check all $s_i'$

do we just fix a particular probability distribution and compute the bayesian NE?

4:56 AM

You could check all $s_i$'s or you could perform this as a maximization exercise (maximizing $i$'s utility by choosing $s_i(\theta_i)$ subject to other players' best responses).

If you mean to "compute Bayesian Nash equilibrium" using numerical methods, then you'd have to be more precise about the problem at hand

Suppose we have the prisoner's dilemma game as a bayesian game

user image

You should read Example 8.E.1 in the MWG textbook. It's exactly about the prisoner's dilemma with incomplete information.

with $T>R>P>S$ and the types could simply be $\Theta_i=\{a,b\}$ such that instead of $R$ as a utility we have $R+a$ or $R+b$.

okay let me check the textbook!

okay the example is about pure strategies

what if we are interested in mixed strategies? since the game is finite we know that there exists at least one equilibrium in mixed strategies

knowing the utilities in the prisoner's dilemma, actions, types and how types affect the utilities, we should be able to compute the bayesian nash equilibrium! going back to your point about the maximization exercise

5:11 AM

For mixed strategy Bayesian Nash equilibrium, you can exploit the necessary condition for such equilibria that a player must be indifferent between all pure strategies in the support of the equilibrium mixed strategy

user image

do you mean this?

Yes, precisely

I thought that this result holds only for normal form games

do we have to transform the bayesian game to a normal form game using Selten's way?

No, it's true generally.

I see! that's great

5:15 AM

You could do the Selten transformation as well

Incomplete info games are essentially games with one extra player (Nature) who always randomizes

and there is no need to fix the probability distribution?

to my understanding we have the common prior assumption and then the probability distributions of the belief functions and then the mixed strategies

we can analyse a game without ever fixing the probability distributions, is that right?

Common prior simply means that Nature's mixed strategy is known

fixing which prob distribution?

probability distribution over actions? or over states?

I haven't studied the bayesian games with the states

so, over the actions

we also have the belief functions! each player is uncertain about the other players' types

we define a function $p:\Theta_i\to\Pi(\Theta_{-i})$

Probability distribution over states is usually the common prior. That's usually given/fixed.
Probability distribution over actions is the mixed strategy that you'd hope to solve for. So you should not be fixing these

are the states equivalent to the types of the players?

5:23 AM

Yes, just different names in different applications

okay!

yes, so what do we do with the belief functions?

Beliefs are tricky to explain, especially over chat.

fair enough

I'd suggest that you read Section 9.C of MWG

do you mind if we keep this discussion open? I want to work on this a little bit more and ask you for verification. if you're willing to take the time that is!

5:25 AM

Sure. But I'll have to go to bed soon. I can answer your questions tomorrow morning though.

yes of course! i'll work on this material for the next few days.

thanks for taking the time! much appreciated!

You're welcome. And thanks for the mathjax bookmark. It seems to work well!

17 hours later…

10:50 PM

Hey, I was doing some reading today and I found that the beliefs don't need to be tricky after all!

We can assume that the beliefs of the different players are posteriors, obtained by conditioning a common prior on the player's type.

since the common prior is fixed in most cases, we can expect the players to compute the posteriors using the Bayes

Bayes' rule...

although one may say that this is a pretty strong assumption!

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Transcript for

Feb '197

Discussion between johnny09 and Herr K.

Imported from a comment discussion on economics.stackexchange....

bayesian-game game-theory

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