1. Christos H. Papadimitriou UC Berkeley www.cs.berkeley.edu/~christos
Fun with Games
Christos H. Papadimitriou
UC Berkeley

2. (Mathematical tools: combinatorics, logic)
Goal of CS Theory ( ):
Develop a mathematical understanding of the capabilities and limitations of the von Neumann computer and its software –the dominant and most novel computational artifacts of that time
(Mathematical tools: combinatorics, logic)
What should Theory’s goals be today?
fun, May 29, 2001

3. fun, May 29, 2001

4. The Internet huge, growing, open, anarchic
as information repository: huge, available, unstructured
built, operated and used by a multitude of diverse economic interests
theoretical understanding urgently needed
fun, May 29, 2001

5. new math for the new Theory?
cf: George Boole The Laws of Thought, 1854
Part I: propositional logic, Part II: probability
cf: John von Neumann The Report on EDVAC, 1945
Theory of Games and Economic Behavior, 1944
(cf: Alan Turing On Computable Numbers, 1936
Studies in Quantum Mechanics, )
fun, May 29, 2001

6. Games, games…
strategies
strategies
3,-2
payoffs
fun, May 29, 2001

7. matching pennies
prisoner’s dilemma
1,-1
-1,1
3,3
0,4
4,0
1,1
auction
chicken
1 … n 1 . n
0,0
0,1
1,0
-1,-1
0, v – y
u – x, 0
fun, May 29, 2001

8. concepts of rationality
undominated strategy
(problem: too weak)
dominating srategy
(problem: too strong, rarely exists)
Nash equilibrium
(problem: may not exist)
randomized Nash equilibrium (  P?)
Theorem [Nash 1952]: Always exists. .
fun, May 29, 2001

9. if a digraph with all in-degrees 1 has a source,
then it must have a sink
 Sperner’s Lemma
 Brouwer’s Fixpoint Theorem
 Kakutani’s Theorem
Nash’s Theorem
min-max theorem
linear programming duality
fun, May 29, 2001

10. More problems: Too many Nash equilibria
Also, Nash equilibria may be “politically incorrect:” Prisoner’s dilemma
Repeated prisoner’s dilemma? PD n
Herb Simon (1969): Bounded Rationality
“the assumption that reasoning and
computation are infinitely cheap
is often at the root of negative results in Economics”
Idea: Prisoner’s dilemma played by memory-limited players?
fun, May 29, 2001

11. c d d c d c d c c c, d c punish once tit-for-tat d d c d c,d c c d c c
punish forever
switch on d
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12. e.g., pricing multicasts [Feigenbaum, P., Shenker, STOC2000]52 30
costs {} 21 21 40 70
{11, 10, 9, 9}
{14, 8}
{9, 5, 5, 3} 32
{23, 17, 14, 9}
{17, 10}
utilities of agents in the node
(u = the intrinsic value of the information
to agent i, known only to agent i) i
fun, May 29, 2001

13. We wish to design a protocol that will result in the computation of:
x (= 0 or 1, will i get it?)
v (how much will i pay? (0 if xi = 0) )
protocol must obey a set of desiderata: i i
fun, May 29, 2001

14. strategy proofness: (w = u  x  v ) w (u …u …u )  w (u … u'…u )
0  v  u,
lim x = 1
strategy proofness: (w = u  x  v )
w (u …u …u )  w (u … u'…u )
welfare maximization
 ui xi – c[T] = max
marginal cost mechanism i i i
u  i
def i i i i i 1 i n i 1 i n
budget balance
 v = c ( T [x])
Shapley mechanism i i
fun, May 29, 2001

15. our contribution:
In the context of the Internet, there is another desideratum:
Tractability: the protocol should require few
(constant? logarithmic?) messages per link.
This new requirement changes drastically the space
of available solutions.
fun, May 29, 2001

16. Shapley mechanism 0  v  u lim x = 1
strategy proofness: (w = u  x  v )
w (u …u …u )  w (u … u'…u )
welfare maximization
 w = max
marginal cost mechanism i i i
u  i
def i i i i i 1 i n i 1 i n
budget balance
 v = c ( T [x])
Shapley mechanism i i
fun, May 29, 2001

17. Games: Fun Deep Useful The mathematical tool of choice of the new TCS
fun, May 29, 2001

18. Bounding Nash equilibria: the price of anarchy
cost of worst Nash equilibrium
“socially optimum” cost s t
3/2 [Koutsoupias and P, 1998]
general
multicommodity
network
2 [Roughgarden and Tardos, 2000]
fun, May 29, 2001