Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can...
Transcript of Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can...
![Page 1: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/1.jpg)
Multiple Equilibria
Some natural games, can have more than one Nash equilibrium, and in this case it becomes
difficult to predict how rational players will actually behave in the game.
![Page 2: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/2.jpg)
A Coordination Game
When two players’ shared goal is really to
coordinate on the same strategy.
![Page 3: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/3.jpg)
PowerPoint VS
Apple’s Keynote software
![Page 4: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/4.jpg)
You both have the same goal, Either would be fine, but
it will be much easier to merge your slides together with your partner’s if you use the same
software. If you both use the same software the payoffs will be
1.
![Page 5: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/5.jpg)
דוגמאות נוספות
• חברות יצור שעובדות יחד צריכות לבחור ביחידות
מידה
• יחידות בצבא צריכות להחליט מאיפה לתקוף
• שני אנשים שהלכו לאיבוד צריכים להחליט מאיפה
להתחיל לחפש או לחלופין אם להישאר במקום או
לזוז
• בכל המקרים האלה יש שתי אפשריות לשווי משקל
? אז באיזה דרך נבחר
![Page 6: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/6.jpg)
The focal point
• is a solution that people will tend to use in the absence of communication, because it seems natural, special or relevant to them
• The concept was introduced by the Nobel Prize winning American economist Thomas Schelling in his book The Strategy of Conflict (1960).
![Page 7: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/7.jpg)
The four squares game
??איזה ריבוע תבחר
רוב האנשים יבחרו באדום
![Page 8: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/8.jpg)
Which side to go???
suppose two drivers are approaching each other at night on an undivided country road. Each driver has
to decide whether to move over to the left or the right.
![Page 9: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/9.jpg)
Variants on the Basic Coordination Game
• אם אתה מגלה פתאום שהמרצה שלך מעדיף את
אחת התוכנות על פני השניה
• עדיין יש שני שווי משקל אבל הם לא שווים
![Page 10: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/10.jpg)
כאשר אתה מעדיף קינוט והשותף שלך מעדיף
פאור פוינט
![Page 11: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/11.jpg)
,שני אנשים הלכו לאיבוד בקניון
האם הם יתחילו לחפש אחד את השני מכיוון
?צפון או דרום
:כעת נוסיף למשחק את הנתון הבא
בכיוון צפון יש חנות ששניהם אוהבים
, על פי תומס שלינג הם יתחילו לחפש מכיון צפון. בגלל ששם החנות שהם אוהבים
![Page 12: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/12.jpg)
Battle of the Sexes
![Page 13: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/13.jpg)
,בעל ואישה רוצים ללכת לסרט
?האם ילכו לסרט רומנטי או סרט מתח
• נותן רווח גדול יותר ( רומנטי, רומנטי)הזוג הסדור
לאישה
• נותן רווח גדול יותר ( מתח, מתח)אך הזוג הסדור לבעל
![Page 14: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/14.jpg)
one final variation on the basic Coordination Game:
Stag Hunt Game
4אם הם ינסו לצוד צבי ביחד הם יצליחו לתפוס אותו ולהרוויח , שני אנשים יצאו לציד
.נקודות
ובזמן הזה הצייד השני עדיין , אם אחד מהם ינסה לצוד את הצבי לבדו הוא לא יצליח
.נקודות 3יכול לתפוס ארנבת ולהרוויח
נקודות מסתכן בלהפסיד הכול 4כאן ניתן לראות שהצייד שינסה להרוויח
.נקודות 3הצייד שיבחר ישר בארנבת יצליח להרוויח , לעומת זאת
![Page 15: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/15.jpg)
דילמת האסיר
• המשטרה עצרה שני עבריינים שביצעו :דילמת האסיר
אם תצליח . ומפרידה ביניהם לצורך חקירה, פשע משותף
-ייכנס כל אחד מהם לכלא ל, המשטרה להביא להרשעתם
אך בחוסר ראיות הם יועמדו לדין על עבירה , שנה 15
. משנית שבגינה ייכנס כל אחד מהם לכלא לשנה אחת
ולכן היא מציעה , למשטרה אין די ראיות להעמידם לדין
וכפרס מובטח לעד עונש , לכל אחד מהם להעיד נגד רעהו
, אם שני האסירים יקבלו את הצעת המשטרה: מופחת
ואם רק אחד מהם , ייכנס כל אחד מהם לכלא לחמש שנים
-העד יצא מיד לחופשי וחברו ייכלא ל, יעיד ורעהו ישתוקשנה 15
![Page 16: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/16.jpg)
Prisoner’s Dilemma
יחליט לסמוך על חברו 1' אם אסיר מס, באותו האופן בדילמת האסיר
הוא מסתכן בכך שחברו לא יסמוך עליו , וינסה לקבל אפס שנים בכלא
לעומת זאת אם הוא יחליט להלשין על . שנה 15והוא יכנס לכלא ל.שנים בכלא 5חברו הוא יקבל מקסימום
![Page 17: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/17.jpg)
Exam-or-Presentation Game
• 84אם רק אחד עושה את המצגת הציון הוא
![Page 18: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/18.jpg)
The Hawk-Dove Game Suppose two animals are engaged in a contest to decide how a
piece of food will be divided between them. Each animal can choose to behave
aggressively (the Hawk strategy) or passively (the Dove strategy). If the two animals both behave
passively, they divide the food evenly, and each get a payoff of 3. If one behaves
aggressively while the other behaves passively, then the aggressor gets most of the food, obtaining a
payoff of 5, while the passive one only gets a payoff of 1. But if both animals behave
aggressively, then they destroy the food (and possibly injure each other), each getting a payoff of 0.
![Page 19: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/19.jpg)
![Page 20: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/20.jpg)
:מבחן-נחזור למשחק מצגת
60אם אף אחד מכם לא עובד על המצגת תקבלו ציון על המצגת
![Page 21: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/21.jpg)
“attack-defense” games
• In such games, one player behaves as the attacker, while the other behaves as the defender.
• The attacker can use one of two strategies let’s call them A and B
while the defender’s two
strategies are “defend against A”
or “defend against B.”
![Page 22: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/22.jpg)
A simple attack-defense game is called Matching Pennies
![Page 23: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/23.jpg)
Mixed Strategies
We will refer to some game as mixed strategies, since they involve “mixing” between the options H and T.
![Page 24: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/24.jpg)
Payoffs from Mixed Strategies
• each player will get +1 with some probability, and will get −1 with the remaining probability
• In order to rank random payoffs numerically, we will attach a number to each distributionthat represents how attractive this distribution is to the player.
• Once we have done this, we can then rank outcomes according to their associated number.
![Page 25: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/25.jpg)
if Player 1 chooses the pure strategy H while Player 2 chooses a probability of q, as above, then the expected payoff to Player 1 is:
• (−1)(q) + (1)(1 − q) = 1 − 2q.
Similarly, if Player 1 chooses the pure strategy T while Player 2 chooses a probability of q,
then the expected payoff to Player 1 is
or (1)(q) + (−1)(1 − q) = 2q − 1.
![Page 26: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/26.jpg)
Equilibrium with Mixed Strategies
• If Player 1 believes that Player 2 will play H strictly more than half the time, then she should definitely play T
• So the point is that the choice of q = 1/2 by Player 2 makes Player 1 indifferent between playing H or T
• each player should randomize so as to make the other player indifferent between their two alternatives. This way, neither player’s behavior can be exploited by a pure strategy, and the two choices of probabilities are best responses to each other.
![Page 27: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/27.jpg)
interpret mixed-strategy in real-world situations.
• a tennis player may be randomly deciding whether to serve the ball up the center or out to the side of the court.
• a card-player may be randomly deciding whether to bluff or not.
• two children may be randomizing among rock, paper, and scissors
![Page 28: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/28.jpg)
Mixed Strategies: Examples and Empirical Analysis
• Because mixed-strategy equilibrium is a subtle concept, it’s useful to think about it through further examples. We will focus on two main examples.
![Page 29: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/29.jpg)
The Run-Pass Game
• The offense can choose either to run or to pass, and the defense can choose either to defend against the run or to defend against the pass.
![Page 30: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/30.jpg)
• Suppose the defense chooses a probability of q for defending against the pass.
• The offense passes
Then the payoff to the offense:
(0)(q) + (10)(1 − q) = 10 − 10q
The offense runs
Then the payoff to the offense:
(5)(q) + (0)(1 − q) = 5q.
• To make the offense indifferent between its two strategies, we need to set 10−10q = 5q,
and hence q = 2/3.
![Page 31: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/31.jpg)
• suppose the offense chooses a probability of p for passing
• The defense defense against the pass
Then the expected payoff to the defense:
(0)(p) + (−5)(1 − p) = 5p − 5
The defense defense against the runs
Then the expected payoff to the defense:
(−10)(p) + (0)(1 − p) = −10p.
• To make the defense indifferent between its two strategies, we need to set 5p−5 = −10p,
and hence p = 1/3.
![Page 32: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/32.jpg)
The Penalty-Kick Game.
• Two-player, two-strategy games
• The kicker can aim the ball to the left or the right of the goal.
• The goalie can dive to either the left or right as well.
• Based on these decisions the kicker is likely to score or not.
![Page 33: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/33.jpg)
• Using the principle of indifference as in previous examples, we see that if q is the probability that a goalie chooses L, we need to set q so as to make the kicker indifferent between his two options:
(.58)(q) + (.95)(1 − q) = (.93)(q) + (.70)(1 − q).
q = .42 and p = .39.
![Page 34: Multiple Equilibriaatcs131/wiki.files/Adi-Multiple...Multiple Equilibria Some natural games, can have more than one Nash equilibrium, and in this case it becomes difficult to predict](https://reader033.fdocuments.in/reader033/viewer/2022050609/5faff6f8e69b14023645413c/html5/thumbnails/34.jpg)
פירוט החישוב:
q ההסתברות שהשוער בועט שמאלה-
1-q השוער בועט ימינה
הבועט בועט שמאלה q0.58+(1-q)0.95= הבועט בועט ימינה
q0.93+(1-q)0.7
p הבועט בועט שמאלה
1-p הבועט בועט ימינה
השוער קופץ ימינה
-0.58p+(1-p)-0.93= השוער קופץ ימינה
-0.95p + (1-p)-0.73