Game Theory VL
Lehrveranstaltung UZH WWF (6 Credits), 2013
Lehrveranstaltung UZH WWF (6 Credits), 2013
Kartei Details
| Karten | 67 |
|---|---|
| Sprache | Deutsch |
| Kategorie | VWL |
| Stufe | Universität |
| Erstellt / Aktualisiert | 21.02.2013 / 22.07.2020 |
| Weblink |
https://card2brain.ch/box/game_theory_vl
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| Einbinden |
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What does it mean for the NE, if the ranking of the payoffs of 2 games is the same?
--> they have the same PNE
What do we need in 2 games to have the same MNE?
we need a positive affine transformation with
u1 (Game2) = alpha x u1(Game1) + Beta
for both players!!
a) how many pure strategies do the players have?
b) find all the subgames
a) P1: 4 info sets, 2 possible actions -->
2^4 = 16 strategies
player 2: 3 info sets, 2 possible actions -->
2^3 = 8strategies
b) 5 subgames
What is a strategy in an extensive form game?
it tells a player what to do at each info set that may be reached (= complete plan of action)
Consider the two game trees.
(a) There is something wrong with the game tree on the right hand side. What?
At P3's infoset, P3 cannot distinguish whether P2 played a or b. Therefore the possible actions at both nodes must be the same (f.e. T and B at both nodes)
What's the difference between a SPE and a NE?
In the SPE, each player's strategy is required to be optimal for every history after it is the player's turn to move, not only at the start of the game, as in a NE.
prisoner's dilemma:
How many possible histories after t = 2?
1 x 2 x 2 x 2 x 2 =16
prisoner's dilema:
How many strategies when T = 2?
5 nodes for each player with 2 possible actions:
2^5 = 32 strategies
what's the subgame perfect equilibrium in repeated games with finite T?
if the normal form game has only 1 NE:
--> The NE is played in every stage!
if the normal form game has multiple NE:
--> game has to end with one of the NE (glaubi!?)
& find delta: ..... next card ;)
Is it a NE?
Yes, if:
5 + 4 x delta > 6 + 1 x delta ==> if delta > 1/3
SPE in repeated games with infinite T:
How to find delta (GRIM strategy) ?
cooperation > deviation:
C-Profit x 1 / (1-delta) > C-profit + D-profit x delta / (1-delta)
What does the Folk Theorem say?
in repeated games, if agents are patient enough (delta is high enough) and T --> unendlich,
many average payoffs (above the NE payoffs) can be attained with SPE strategies.
A Static Bayesian Game consists of:
...?
- The set of players I
- For each player i:
a set of possible types T.i & a set of actions A.i
- payoffs which depend on the actions taken by all players and types
- Probability measure on types p: T --> (0, 1)
- All these elements are common knowledge
How to solve a game with incomplete information!?
--> What was John Harsanyi's great idea?
To transform a game of incomplete information into a game with imperfect information.
To do so he introduced a new player, nature, who moves at the beginning
Definition of a Bayesian Nash Equilibrium!?
In a static Bayesian Game the strategies s* form a (pure-) strategy bayesian NE if for each player i and for each of i's types t.i no player wants to change her strategy, even if the change involves only one action by one type.
Give some examples for games with incomplete information. !?
- auction: Bidders do not know how much the other bidders value the object
- an insurance company does not know for sure the characteristics of the client
- we do not know if a second hand car is good or not
- a country does not know the exact military strenght of the enemy
- entrant / incumbent game
- etc....
If mü is the belief of player 2: probability that he is at the left note, given that he is at his information set.
How do we find mü?
--> picture
a) with what probability is P2's info set reached?
b) with what probability is p2's left node reached? and the right one?
c) What belief mü should player 2 have?
d) given those beliefs, which one is p2's optimal strategy?
a) 2/3
b) left: 2/3, right: 0
c) mü = 1
d) always play L
Assume now that P1 plays a mixed strategy which attaches equal probability to each of her pure strategies. Given this stragey for p1,
a) With what prob is P2's info set reached?
b) prob for left & right node?
c) mü?
a) 5/6
b) left node: 2/3, right node: 1/6
c) mü = 4/5
....give those beliefs, wich one is p2's optimal strategy?
b) is that strategy, together with p1's strategy, a NE?
c) expected value of L? and of R?
a) ????
b) ????
c) L: 4 x (4/5) + 0 x (1/5) = 16/ 5
R: 3 x (4/5) + (5/5) = 17 / 5
--> R > L
What is the unique solution to all bargaining problems?
The solution, that satisfies Nash's four axioms!
If...
- "you do not leave anything on the table"
- the outcome doesn't depend on the utility function
- one doesn't gets more than the other one
- Assume that you think (x.y) should be the solution and that I tell you that a set of partitions cannot be chosen but (x,y) is still possible. --> you still want (x,y) to be the solution!
What are Nash's four axioms, that lead to the Nash bargaining solution?
1.) Invariance to equivalent utility representations (If we change the utilities according to positive affine transformation, the solution should not change!)
2.) pareto efficiency
3.) symmetry (if the bargaining problem is symmetric, then both players should be treated the same)
4.) Independance of irrelevant alternatives (options which are not selected should not change the solution of they are no longer available)
To what do this four nash axioms lead?
(Formel für Problemlösung im bargaining)
The solution assigns to the bargaining problem the pair of payoffs that solves the problem:
R / F ?
a) If we make on of the player more risk averse (make his utility more concave) the bargaining problem changes and the less risk averse bargainer gets a higher share.
b) Improving a bargainer's outside option will make him better off in the bargaining
a) true
b) true
Describe the Rubinstein's Bargaining Game
P1 & P2 play:
- a cake of size one has to be divided. In a starting round, P1 makes a proposal tha P2 can accept or turn down
- if he accepts the cake is split according to this
- if he declines the games moves one round forward and it is now P2 who proposes a share
---> This game can last forever, but time is valuable!!
payoff z, after n round is worth:
delta ^n x Z
(set of Nash equilibria is infinitely large in this game!)
What's rubinsteins bargaining solution?
we solve first for a finite number of periods --> find the SPNE
--> it can be shown that for infinite T, there is a unique SPNE in which:
1 gets 1/ (1+ delta) and 2 gets: delta / (1+delta)
Compare the Nash bargaining solution with the NE in general.
Does the NE fulfill the 4 axioms of the bargaining solution?
1.) Invariance to equ. utility representations: YES
(NE are invariant to affine (=equivalent) utility representations)
2.) pareto efficiency: NO
3.) Symmetry: NO
4.) Independance of irrelevant alternatives: ??
