An introduction to game theory: strategies, game equilibria (min-max, Nash), cooperative games and Shaply value.

1. Introduction to game
theory
Nadav Carmel
11-10-2018

2. outline
 Introduction + basics
 Pure vs. mixed strategies
 Zero sum games (constant sum game)
 Definition
 Example
 Maximal confidence strategy
 Min-Max theorem – John von Neumann
 Non-zero sum game
 Example: prisoner dilemma (pure strategies)
 Example: battle of sexes
 Pure strategies
 Mixed strategies
 Definition: dominant + dominated strategies
 Nash theorem - John Forbes Nash
 Stable marriage problem - David Gale and Lloyd Shapley
 Problem definition
 Application
 Example
 Solution (algorithm)
 Coalitional (cooperative) games
 Definition
 Shapley Value
 Auctions
 Cool vid 

3. Game Theory – basic facts
 Game theory is the study of mathematical models of strategic interaction
between rational decision-makers.
 It has applications in all fields of social science, as well as in logic and
computer science.
 The modern game theory developed in the 20th century, with the significant
contributions of:
 John von Neumann (game definition, zero sum game equilibrium, minmax theorem)
 John Forbes Nash (Nash equilibrium theorem)
 David Gale and Lloyd Shapley (stable marriage problem, Shapley Value)
 Cold war – significant US research investment

4. What is a game
 A game is a model for a conflict between players.
 For each player, a number of strategies (actions) can be taken.
 The actions take place simultaneously.
 Given the chosen strategies, the result is a pair of numbers, representing the
payoff for each player (value).

5. Pure vs mixed strategies
 A pure strategy provides a complete definition of how a player will play a
game. In particular, it determines the move a player will make for any
situation he or she could face. A player's strategy set is the set of pure
strategies available to that player.
 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.

6. Definitions - dominant strategies
 B strictly dominates A: choosing B always gives a better outcome than
choosing A, no matter what the other player(s) do.
 B weakly dominates A: There is at least one set of opponents' action for
which B is superior, and all other sets of opponents' actions give B the same
payoff as A.
 B is weakly dominated by A: There is at least one set of opponents' actions
for which B gives a worse outcome than A, while all other sets of opponents'
actions give A the same payoff as B. (Strategy A weakly dominates B).
 B is strictly dominated by A: choosing B always gives a worse outcome than
choosing A, no matter what the other player(s) do. (Strategy A strictly
dominates B).
 B and A are intransitive: B neither dominates, nor is dominated by, A.
Choosing A is better in some cases, while choosing B is better in other cases,
depending on exactly how the opponent chooses to play

7. Zero sum game
 A zero sum game is a game where the players interests are completely
opposite.
 Formally, the payoff of one player equals the minus payoff of the other.

8. Zero sum games – example
 Odd-Even game
 Assume player 1 is Even, and player 2 is Odd
 Payoff matrix is:
Player 1 (E) Player 2 (E) Odd Even
Odd 1, -1 -1, 1
Even -1, 1 1, -1

9. Zero sum game equilibrium
 Look again at our zero sum game:
 Player 1 whishes to maximize his payoff, what is his optimal strategy?
 If player 2 had Odd, then Even is the optimal strategy.
 Otherwise, Odd is the optimal strategy.
 Optimal strategy of player 2 depends on the outcome of player 1.
 If only pure strategies are allowed for the game, optimal strategies are cyclic
→ No equilibrium exists here.
Player 1 (E) Player 2 (E) Odd Even
Odd 1, -1 -1, 1
Even -1, 1 1, -1

10. Definition – Maximal confidence strategy
 A strategy will be called maximal confidence if for ‘worst possible’ strategy of
player 2, player 1 has the optimal return:
𝑖0 = max
𝑖
min
𝑗
𝑈 𝑖, 𝑗
 Lemma: for any payoff matrix:
max
𝑖
min
𝑗
𝑈 𝑖, 𝑗 ≥ min
𝑖
max
𝑗
𝑈 𝑖, 𝑗
 A game will have an equilibrium iff: max
𝑖
min
𝑗
𝑈 𝑖, 𝑗 = min
𝑖
max
𝑗
𝑈 𝑖, 𝑗 . This
value will be called the game value.

11. Minmax theorem – John von Neumann
 The minimax theorem is a theorem providing conditions that guarantee that
the min-max-inequality is also an equality.
 The theorem is von Neumann's theorem from 1928, which was considered the
starting point of modern game theory.

12. Minmax theorem
 Any zero sum game with mixed strategies has an equilibrium.
 In general:
 The minmax theorem states that for any zero-sum game with mixed
strategies, there exists an equality:
 𝑝 ∈ ∆ 𝑠 𝑖
− 𝑡ℎ𝑒 𝑚𝑖𝑥𝑒𝑑 𝑠𝑡𝑟𝑎𝑡𝑒𝑔𝑖𝑒𝑠 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑛 𝑡ℎ𝑒 𝑠𝑖𝑚𝑝𝑙𝑒𝑥 ∆ 𝑠 𝑖

13. Non zero sum game – example
 Prisoner dilemma:
 In this case, pure strategy equilibrium exists
Equilibrium

14. Non zero sum game – example 2
 Battle of sexes:
 If no agreement is reached – the payoff is zero, otherwise, payoff is positive.
 In this case – 2 pure-strategy equilibria exists.
 Each equilibrium is unfair – one player does better than the other.
Husband wife Football Opera
Football 3,1 0,0
Opera 0,0 1,3

15. Pure vs mixed strategies
 What if we could randomize our strategy selection?
 Payoff for wife choosing Football = p*1 + (1-p)*0
 Payoff for wife choosing Opera = p*0 + (1-p)*3
 In equilibrium, both strategies have equal payoffs → p=3/4
 Similarly for Husband, in equilibrium, q=3/4
 Payoff for wife = payoff for husband = ¾ (less than the return one would
receive from constantly going with the other favored event)
Husband wife Football (q) Opera (1-q)
Football (p) 3,1 0,0
Opera (1-p) 0,0 1,3

16. Nash equilibrium – John Forbes Nash
 The theorem states that for any finite, non-cooperative, game there must be
an equilibrium if mixed strategy is allowed (not necessarily zero-sum games).
 Published a series of papers on this subject during the early 50’s
 Won the 1994 economics Nobel-prize for his theorem.
 Made famous in ‘Beautiful Mind’

17. Nash theorem
 Definition: For strategy S by player 1 and T by player 2, the pair (S,T) is a
Nash equilibrium if S is a best response (dominates all other strategies) to T,
and T is a best response (dominates all other strategies) to S.
 More generally, at Nash equilibrium no player wants to unilaterally deviate to
an alternative strategy:
 𝑆𝑖: arbitrary strategiy of player 𝑖.
 𝑆𝑖
∗
: equilibrium strategiy of player 𝑖.
 𝑆−𝑖: arbitrary strategiy of all players but 𝑖.
 𝑆−𝑖
∗
: equilibrium strategiy of all players but 𝑖.
 If no equilibrium exists for pure strategy, one must exist for mixed strategy.
∀𝑆𝑖: 𝑝𝑎𝑦𝑜𝑓𝑓 𝑆𝑖
∗
, 𝑆−𝑖
∗
≥ 𝑝𝑎𝑦𝑜𝑓𝑓 𝑆𝑖, 𝑆−𝑖
∗

18. Stable marriage problem - Gale–Shapley
 Lloyd Shapley, Nobel Prize Winner 2012 in economics
 Obtained the prize for a number of contributions, one being the Gale-Shapley
algorithm (1962), discussed today

19. The stable marriage problem
 Story: there are n men and n women, which are unmarried. Each has a
preference list on the persons of the opposite sex
 Does there exist and can we find a stable matching (stable marriage): a
matching of men and women, such that there is no pair of a man and a
woman who both prefer each other above their partner in the matching?

20. The stable marriage problem –
application
 Origin: assignment of medical students to hospitals (for internships)
 Students list hospitals in order of preference
 Hospitals list students in order of preference

21. The stable marriage problem –example
 Matching preferences lists:
 Arie: Betty Ann Cindy
 Bert: Ann Cindy Betty
 Carl: Ann Cindy Betty
 Ann: Bert Arie Carl
 Betty: Arie Carl Bert
 Cindy: Bert Arie Carl
 Stable matching: (Arie,Betty), (Bert,Ann), (Carl,Cindy)

22. The stable marriage problem – remark
 “Local search” approach does not need to terminate
SOAP-SERIES-ALGORITHM
While there is a blocking pair
Do Switch the blocking pair
 Can be extremely slow!
(NP-hard. ‘Parameterized Complexity and Local Search Approaches for the Stable Marriage Problem with Ties, 2009’)
 So, we need something else…
 (a blocking pair: a pair which both prefer to be with each other, rather than their current partners)

23. The stable marriage problem - algo
 Gale/Shapley algorithm: finds always a stable matching
 Input: list of men, women, and their preference list
 Output: stable matchings

24. The stable marriage problem - algo
 Fix some ordering on the men
 Repeat until everyone is matched
 Let X be the first unmatched man in the ordering
 Find woman Y such that Y is the most desirable woman
in X’s list such that Y is unmatched, or Y is currently
matched to a Z and X is more preferable to Y than Z.
 Match X and Y; possible this turns Z to be unmatched
Questions:
Does this terminate? How fast?
Does this give a stable matching?

25. The stable marriage problem – algo
complexity
 Termination and number of steps:
 Once a woman is matched, she stays matched (her partner can change).
 When the partner of a woman changes, this is to a more preferable partner
for her: at most n – 1 times.
 Every step, either an unmatched woman becomes matched, or a matched
woman changes partner: at most n2 steps.

26. The stable marriage problem – algo
stability
 Suppose final matching is not stable.
 For example, suppose Arie and Betty who are dissatisfied with the matching.
 Take:
 Arie is matched to Ann
 Arie: Betty > Ann
 Betty is matched to Bert
 Betty: Arie > Bert
 This is impossible, because, if Arie prefers Betty, it means he has asked Betty
and was rejected, meaning, Betty: Bert > Arie, which is a contradiction for
our initial assumption.

27. The stable marriage problem -
comments
 A stable matching exists and can be found in
polynomial time.
 Controversy: the algorithm is optimal for men but
not (necessarily) for woman…

28. Coalitional Games (cooperative games) –
Definition
 Cooperative game theory assumes that groups of players, called coalitions,
are the primary units of decision-making, and may enforce cooperative
behavior.
 Consequently, cooperative games can be seen as a competition between
coalitions of players, rather than between individual players.

29. Coalitional Games (cooperative games)
 Coalition formation is a key part of negotiation between self-interested
players.
 Examples:
 Truck delivery companies can share truck space, as the cost is mostly
dependent on the distance rather than on the weight carried
 Several of companies can unite into a virtual organization to take more
diverse orders and gain more profit

30. Coalitional Games – Player Types
 Dummy players add nothing to all coalitions:
 Equivalent players add the same to any coalition that does not contain any of
the two players:
( { }) ( )v S a v S 
1 2 1 1: { , } ( { }) ( { })S S a a v S a v S a      

31. Coalitional Games –gains distribution
 Given a coalitional game we want to find the
distribution of the gains of the coalition between
the players
 Different solution concepts have different
objectives
 Game theory has studied these solution concepts
for quite some time, but the computational
aspect has received little attention

32. Coalitional Games – The Shapley Value
 Aims to distribute the gains in a fair manner.
 A value division that conforms to the set of the
following axioms:
 Dummy players get nothing
 Equivalent players get the same
 If a game 𝑣 can be decomposed into two sub games, a
player 𝑎 gets the sum of values in the two games:
 Only one such value division scheme exists!
: ( ) ( ) ( )
: ( ) ( ) ( )V U W
S v S u S w S
a d a d a d a
   
  

33. Computing the Shapley Value
 Let 𝐴 be the set of players
 Given an ordering 𝜋 of the players in 𝐴 (permutation), we define 𝑆 𝜋, 𝑎 to be
the set of players of 𝐴 that appear before 𝑎 in 𝜋.
 The Shapley value is defined as the marginal contribution of an agent to its
set of predecessors, averaged on all possible permutations of the agents:
1
( , ) ( ( ( , ) ) ( ( , )))
!
Sh A a v S a a v S a
A 
    

34. A Simple Way to Compute The Shapley
Value
 Simply go over all the possible permutations of the agents and get the
marginal contribution of the agent, sum these up, and divide by |A|!
 Extremely slow
 We can use the fact that a game may be decomposed to sub games, each
concerning only a few of the agents, for computational complexity reduction.

35. Shapley value calculation – example
 The gloves game: a coalitional game where the players have left and right hand gloves
and the goal is to form pairs.
 For example:
 players L have a left glove, and R1 and R2 have right gloves.
 The payoff for a pair of (opposite) gloves is 100, otherwise 0.
 Lets calculate Shapley value per each player:
 𝐴 ! = 3! = 6
 The permutations (𝜋) table is:
 sum_payoff(L) = 400 → payoff(L) = 400/6 = 66.66
 sum_payoff(R) = 100 → payoff(R) = 100/6 = 16.66
 Total coalition payoff = 66.66 + 16.66 * 2 = 100 (check)
1
( , ) ( ( ( , ) ) ( ( , )))
!
Sh A a v S a a v S a
A 
    
L=0 R1=0 R2=100
L=0 R2=100 R1=0
R1=0 L=100 R2=0
R1=0 R2=0 L=100
R2=0 L=100 R1=0
R2=0 R1=0 L=100

36. Auctions
 There are traditionally four types of auction that are used for the allocation of a
single item:
 First-price sealed-bid auctions in which bidders place their bid in a sealed envelope and
simultaneously hand them to the auctioneer. The envelopes are opened and the
individual with the highest bid wins, paying the amount bid.
 Second-price sealed-bid auctions (Vickrey auctions) in which bidders place their bid in a
sealed envelope and simultaneously hand them to the auctioneer. The envelopes are
opened and the individual with the highest bid wins, paying a price equal to the second-
highest bid.
 Open ascending-bid auctions (English auctions) in which participants make increasingly
higher bids, each stop bidding when they are not prepared to pay more than the current
highest bid. This continues until no participant is prepared to make a higher bid; the
highest bidder wins the auction at the final amount bid. Sometimes the lot is only
actually sold if the bidding reaches a reserve price set by the seller.
 Open descending-bid auctions (Dutch auctions) in which the price is set by the
auctioneer at a level sufficiently high to deter all bidders, and is progressively lowered
until a bidder is prepared to buy at the current price, winning the auction.

37. Auctions – FPSB example
 Assume an item valued at 1000$.
 Any strategy of bidding less than 1000$ is better than bidding exactly 1000$ (strongly dominates).
 Denote: 𝑃: 𝑚𝑦 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑝𝑎𝑦𝑜𝑓𝑓, 𝑥: 𝑚𝑦 𝑏𝑖𝑑 𝑝𝑟𝑖𝑐𝑒, 𝑦: 𝑚𝑦 𝑟𝑖𝑣𝑎𝑙′
𝑠 𝑏𝑖𝑑 𝑝𝑟𝑖𝑐𝑒
 𝑃 = 𝑝𝑟𝑜𝑏(𝑥 > 𝑦) ∗ (1000 − 𝑥) + 𝑝𝑟𝑜𝑏(𝑥 < 𝑦) ∗ 0 (assuming bidding is free).
 We wish to maximize our expected payoff
 Assuming one rival with no prior knowledge: 𝑦 ~ 𝑈(0, 1000)
 𝑃 =
𝑥−0
1000
∗ 1000 − 𝑥
 Derive and equate to zero to find optimal solution (w.r.t. x):
𝑥 𝑜𝑝𝑡 = 500 → 𝑃𝑜𝑝𝑡 = 250
 In the multiple rivals (i.i.d.) case:
 𝑃 =
𝑥−0
1000
𝑛
∗ 1000 − 𝑥 , 𝑤ℎ𝑒𝑟𝑒 𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑖𝑣𝑎𝑙𝑠
 𝑥 𝑜𝑝𝑡 =
1000∗𝑛
𝑛+1
→ 𝑃𝑜𝑝𝑡 = 1000
𝑛 𝑛
𝑛+1 𝑛+1

38. Real life example…
 https://www.youtube.com/watch?v=p3Uos2fzIJ0

39. Thank you!