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Principles of Game Theory. Lecture 12: Cheap Talk. Administrative. Homework due tomorrow Quiz Thursday Repeated Games – look at chapter 11 in Dixit and the end of Chp 2 in Gibbons No class Sunday, Sept 29 th ; next quiz Oct 6 (week from Sunday). Rational Irrationality.

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principles of game theory

Principles of Game Theory

Lecture 12: Cheap Talk

administrative
Administrative
  • Homework due tomorrow
  • Quiz Thursday
    • Repeated Games – look at chapter 11 in Dixit and the end of Chp 2 in Gibbons
  • No class Sunday, Sept 29th ; next quiz Oct 6 (week from Sunday)
rational irrationality
Rational Irrationality
  • U.S. / U.S.S.R. nuclear deterrence
    • Mutually Assured Destruction (MAD)
      • like Grim Trigger Strategy
    • Proportional Response
      • like Tit-for-Tat
  • Dr Strangelove.
doomsday device
Doomsday device

Is it credible?

  • Severity

“Create fear in the mind of the enemy”

  • Irreversibility

“It is essential”

  • Irrationality

“Not something a sane man would do”

  • Practicality

“It wasn’t a practical deterrent”

  • Clarity

“Tell the world”

commitment under uncertainty
Commitment Under Uncertainty

For you seniors:

  • An offer you can’t refuse:
    • After a seemingly successful interview, the interviewer asks where the firm ranks on your list of potential employees
    • BUT Before answering, you are told:
      • The firm only hires applicants who rank it first
      • If the firm is in fact your first choice, then you must accept a job offer in advance, should one be made
commitment under uncertainty1
Commitment Under Uncertainty

Why make such offers?

  • Take advantage of your uncertainty
  • Take advantage of your risk-aversion
  • Make you commit before they do!
games in real life
Games in real life
  • Last time I asked you to write down an interaction as a game. So let’s hear about them
communication
Communication
  • So far we’ve assumed that the players can’t talk before they play the game
    • At most players could make moves that “signal” their intentions.
  • But people talk all the time before they actually interact. How does that change things?
    • Behaviorally: lots of implications (not necessarily consistent)
    • Theoretically: it depends.
back to simple games
Back to simple games
  • Split into two groups and have a sheet of paper ready
  • Now we’ll allow communication…
game 1
Game 1
  • Group 1: you may, but are not compelled to, write a message to Group 2 before selecting your action
game 2
Game 2
  • Group 2: you may, but are not compelled to, write a message to Group 1 before selecting your action
game 3
Game 3
  • Group 1: you may, but are not compelled to, write a message to Group 2 before selecting your action
any strategic moves
Any strategic moves?
  • Can anyone think of a strategic move group 2 – the receiver – could have made in the last game?
talk is cheap
Talk is cheap…
  • These games are examples of what we call “Cheap Talk” games
    • Sending the message is costless (hence cheap)
    • How do we know that a cheap (costless) message is informative?
      • Short answer is that we don’t…
      • At best it changes the beliefs of the players (note: why both players and not just the receiver?)
equilibria in cheap talk games
Equilibria in Cheap Talk Games
  • We’ll analyze the strategy space and equilibria in more detail next time, but:
    • With perfectly aligned interests (pure coordination – game 1) cheap talk can help with equilibrium selection
    • Similarly with partially aligned interests (game 3)
    • With opposing interests, why should a player believe what the other is saying?
equilibria in cheap talk games1
Equilibria in Cheap Talk Games
  • With communication, there are lots, and LOTS, of equilibria
    • Often “nonsensical” ones are referred to as babbling equilibria
      • It’s clearest to see these in zero-sum games
      • But…they actually still exist in games of perfectly aligned interests as well
        • Much in the same way non-subgame perfect equilibria exist in extensive form games.
bayes theorem
Bayes Theorem

In preparation for next time, I want to make sure you are familiar with Bayes Theorem:

But it’s often useful to use the law of total probability:

bayes theorem1
Bayes Theorem
  • When those A’s and B’s (types of other players and messages that can be sent) are continuous it requires integration over a densities fAand fB:
bayes tables
Bayes Tables

But when there are only a couple of events (types of players, etc), we can construct a table:

where a and b are often described as the accuracy of the test:

  • a = P(“A”|A) or using the notation in the table P(X|A)
  • b = P(“B”|B) or using the notation in the table P(Y|B)
an example
An Example
  • H = Patient HIV+ is either {0,1}
    • Patient is actually either {negative, positive}
  • T = Patient tests positive for HIV+ is either {0,1}
    • Or the test says {“negative, “positive”}
  • What could happen?
    • “True Positive:”P(H = 1 | T = 1) is the probability of being HIV+ given testing positive for HIV+
    • “False Positive:” P(H = 0 | T = 1) is the probability of not being HIV+ given testing positive for HIV+
    • “False Negative”: P(H =1 | T = 0) is the probability of being HIV+ given testing negative for HIV+
    • “True Negative”: P(H=0 | T=0) is the probability of not being HIV+ given testing negative for HIV
an example1
An Example

Assume

    • P(HIV=1) = 0.0076
    • P(“Positive” | HIV = 1 ) = 0.976
    • P(“Negative” | HIV = 0 ) = 0.995
  • What’s the probability of having HIV given the test came back positive? P( HIV | “Positive”)
the probability matrix
The Probability Matrix

1 – Rate of True Positives

(“False Negatives”)

Rate of True Positives

(“Accuracy”)

Joint Probability

0.024 × 0.0076

1 – Rate of True Negatives

(“False Positives”)

Probability Test

Says NOT HIV+

0.0002 + 0.9874

P(HIV|“Positive”) = 0.0074/0.0124 = 0.5992