Introduction to game theory and auctions

1
Introduction to game theory and auctions

• Moshe Tennenholtz
• Technion and Stanford University

2
Outline

• Game theory
• Static games
• Dynamic games
• Games with incomplete information
• Mechanism design
• From analysis to design
• Auctions
• Some results of auction theory

3
Game theory

• Game theory deals with the interaction of several
  self-motivated agents, extending upon classical
  decision theory.
• In decision theory an agent has to take decision
  adopting beliefs about the environment

4
Game theory

• Game theory deals with the situation where
  several agents have to take decisions adopting
  beliefs about one another.
• Battle of the sexes

5
Static games

• In a static game each agent has a set of possible
  strategies to choose from, and a payoff
  function.
• An agents payoff is determined by the strategies
  selected by all agents.
• Pure coordination
  game

6
Static Games in Strategic Form

• A (two-player) game in strategic form is a tuple
  ltS1, S2, U1, U2gt where S1 is a set of strategies
  available to player i, and Ui S1S2?R is a
  utility/payoff function for player i.
• Usually depicted through a payoff matrix

7
Examples of game in strategic form

• Prisoners Dilemma (PD)
• The coordination game
• Matching pennies

8
Larger games

• Larger payoff matrix (as many dimensions as
  players)
• Each dimension has as many entries as there are
  possible strategies (actions) to that agent
  different agents may have different numbers of
  strategies

9
Solving games

• How will agents behave in a game?
• The solution is not obvious and much of game
  theory deals with this subject.
• The basic approach agents will choose an
  equilibrium strategy.
• A joint strategy of the agents is in
  equilibrium if it is irrational for each agent to
  deviate from it assuming the other agents stick
  to their part of that joint strategy

10
Solving games

• Defection by all agents is the only equilibrium
  of the famous prisoners dilemma

11
A solution concept the Nash equilibrium.

• A pair of strategies (s,t) is a Nash equilibrium
  if
• ?(s'? S1, t'? S2), U1(s', t) ? U1(s, t),
  U2(s, t') ? U2(s, t)

12
Solving games

• Equilibrium analysis enables to analyze games.
• In certain cases equilibrium analysis leads to
  paradoxes, which are then studied and leads to
  refinements and improvements of solution
  concepts.
• In certain cases an equilibrium is not unique
  and equilibrium selection becomes a major issue
  (see the trust game below)

13
Strategy Types

• Dominant Strategy
• Best to do no matter what others do
• e.g., prisoners dilemma (PD) has a dominant
  strategy (best to do no matter what others do)
• The coordination game has several equilibria, but
  no dominant one.

14
Mixed Strategies

• Mixed strategies of player i probability
  distributions on Si, denoted by ?(Si).
• The definition of Nash equilibrium is easily
  generalized to mixed strategies rather than look
  at payoff, look at expected payoff.

15
Solving games

• But, does equilibrium always exist?
• YES! (Nash), if we allow mixed
  (probabilistic) strategies.
• Thm. There always exists a Nash equilibrium in
  mixed strategies. The result holds also for the
  case of n players.
• Agent 1 (resp. 2) will give probability 1/3 to
• boxing (resp. concert)
• is an equilibrium of the battle of the sexes.

16
Zero-Sum Games

• Zero-sum games U1-U2
• we will refer only to the payoff of player 1.
• The value of a zero-sum game is the payoff
  obtained by player 1 in equilibrium, and it is
  independent of which equilibrium is selected.

17
Dynamic and multi-stage games

• In a dynamic game there are several stages, and
  an agents strategy may depend on the history of
  his/her actions taken so far.
• An example play the battle of the sexes 100
  times (e.g. once a day). You may decide to choose
  boxing on a particular day if and only if concert
  has been selected by the other agent no more than
  5 times.
• Extensive form games agents may alternate in
  making their moves, e.g. in a revised version of
  the battle of the sexes agent 1 will make his
  move to be followed by agent 2s move.

18
Repeated Games in Strategic Form

• We play the game finitely or infinitely many
  times.
• Strategies may depend on the whole history.
• Example 1 finitely repeated PD
• backward induction.
• Example 2 infinitely repeated PD
• TFT (tit-for-tat).

19
Games in Extensive Form Game Trees

• A two-player game in extensive form is a tree
  where odd levels are associated with player 1,
  and even levels are associated with player 2, and
  the leaves are associated with the players
  payoffs.
• Extensive form games can (in principle) be solved
  by the minimax algorithm (e.g., Chess).
• In the case of non zero-sum games, this procedure
  leads to certain paradoxes.
• Example (over)

20
the centipede game
A
B
A
B
A
B
B
101,99
1,0
0,2
3,1
96,98
99,97
98,100
2,4
21
Nash- vs subgame-perfect- equilibriumin
extensive-form games

• Consider the following game tree
• There are two Nash equilibria (U,R) and (D,L)
• But only one subgame-perfect eqm (U,R)

D
U
1,2
L
R
2,1
0,0
22
Games with incomplete information

• There are several ways to model uncertainty in
  games.
• The most classical way to model uncertainty is
  by referring to partial information about agents
  utility/payoff functions
• The payoff/utility function of an agent is taken
  to depend on a parameter called the agents type,
  which is typically known to the agent but it is
  unknown to the other agents.
• An agents type is for example his willingness to
  pay for a particular good.
• The typical assumption is that the distribution
  on the agents types is known, each agent knows
  his/her type but in general he/she does not know
  the type of other agents.

23
Games with incomplete information

• The equilibrium concept has been generalized to
  games with incomplete information (Harsanyi)
• A joint strategy of the agents is in equilibrium,
  if each agent applies its best response against
  the strategies of the other agents, given the
  distribution on agents types.

24
Uncertainty Bayesian Games

• Represent games in which agents have partial
  information about one another
• Bayesian games add this ingredient in one of two
  equivalent ways
• Posit a set of games, with each player having a
  belief (probability) about which is being played
• Posit a single game with an added player, Nature,
  with each player receiving some signal about
  Natures move.
• Bayes-Nash equilibrium is a generalization of
  Nash equilibrium to this setting.

25
Auction as a Bayesian game

• Players bidders Nature
• Nature chooses valuations for each agent
• Each agents signal is his own valuation.
• Agents strategy mapping from valuations to bids

26
Example simple first-price (high bid) auction

• n risk-neutral agents
• Valuations are real numbers, distributed in the
  range between 0 and 1
• Valuations are independently drawn based on a
  uniform distribution.
• One unit of good
• Agents submit monetary bids
• The good is allocated to the agent who has made
  the highest bid this agent will pay his/her own
  bid.
• An agents type is his/her own valuation (maximal
  willingness to pay).
• An agents strategy determines how much he/she
  will bid as a function of his/her type.
• An agents payoff is determined by whether he/she
  received the good, his/her payment, his/her
  type.
• The equilibrium an agent with valuation v, will
  bid (1-1/n)v.
• Notice that agents cheat about their valuation.
  

27
Computing equilibria brief example

• Setting
• One good, two agents
• The agents valuations are independently drawn
  from the uniform distribution on 0,1
• u(y)y is the utility functions of both agents
• A first-price (FP) auction

28
First approach proving a particular equilibirum

• Assume player 1 plays z, and player 2s strategy
  is b(y)y/2
• If player 1s valuation is x his expected payoff
  is given by
• (note given the y/2 strategy, 1 only wins when
  2s valuation is lt2z)
• This is a quadratic equation with derivative
• is equal to 0 at
• The same analysis is true of player 2
• Therefore b(x)x/2 is the best response to the
  same strategy by the other player, and therefore
  the two players adopting this strategy forms an
  equilibrium

29
Second approach finding an equilibirum

• Well be looking for a continuous symmetric
  increasing equilibrium.
• 1s expected payoff is

30
Second approach (cont.)

• We now look for a value for z which zeros the
  derivative, under the constraint that zb(x)
• Now note that b(x)x/2 is a solution

31
Beyond two-player 1st-price auctions

• More generally, with n bidders and similar
  conditions, the symmetric equilibrium is given
  by

32
Example second-price (Vickrey) auction

• In a second-price auction the good will be sold
  to the agent with the highest bid. He/she will
  pay the second highest bid.
• In equilibrium of a second-price auction, each
  agent submits his/her valuation as his/her bid.
• The following strategy is a dominant strategy
  truth-revealing is optimal regardless of other
  agents behavior.

33
Mechanism design from analysis to synthesis

• Given a description of an environment, i.e the
  information structure with regard to the agents
  valuations, the agents utility functions (e.g.
  whether they are risk-neutral, risk-averse,
  risk-seeking), and an optimization criterion
  (e.g. maximizing revenue, efficient computation
  of certain statistics), find an optimal game, a
  one such that in the equilibrium of which (given
  the information structure) we will obtain optimal
  behavior (given the optimization criteria).
• Example assuming the number of participants, the
  information structure as before (risk-neutral
  participants with valuations drawn independently
  and uniformly from the interval 0,1), find an
  auction procedure that optimizes the sellers
  revenue.

34
Auctions

• Auctions are the most widely-studied economic
  mechanism.
• Auctions refer to arbitrary resource allocation
  problems with self-motivated participants.
• The basic insight of auction theory can be
  generalized and extended to be used in other
  forms of trade.

35
Some Classical Assumptions

• Independent valuations for object(s)
• Free disposal
• No Externalities
• Risk-averse/neutral agents (concave utility
  functions most analysis is for risk neutral
  agents).
• Constant risk attitude

36
Auction Rules

• English/Japanese auction
• Dutch auction
• First-price auction
• Second-price auction
• k-price auctions
• Combinatorial (multi-dimensional) auctions lead
  to hard computational problems, but are more
  expressive
• Multi-round auctions lead to complex equilibrium
  analysis and multiple equilibria

37
Single-unit English auction

• Bidders call ascending prices
• Auction ends
• at a fixed time
• when no more bids
• a combination of these
• Highest bidder pays his bid

38
Multi-unit English auctions

• Different pricing schemes
• lowest accepted (uniform pricing, sometimes
  called Dutch)
• highest rejected (uniform pricing, GVA)
• pay-your-bid (discriminatory pricing)
• Different tie-breaking rules
• quantity
• time bid was placed
• Different restrictions on partial quantities
• allocate smaller quantities at same
  price-per-unit
• all-or-nothing

39
Japanese auction

• Auctioneer calls out ascending prices
• Bidders are initially in, and drop out
  (irrevocably) at certain prices
• Last guy standing gets it at that price

40
Dutch (descending clock) auction

• Auctioneer calls out descending prices
• First bidder to jump in gets the good at that
  price
• With multiple units bidders shout out a quantity
  rather than mine. The clock can continue to
  drop, or reset to any value.

41
Sealed bid auctions

• Each bidder submits a sealed bid
• (Usually) highest bid wins
• Price is
• first price
• second price
• kth price
• Note Can still reveal interesting information
  during auction
• In multiple units similar pricing options as in
  English

42
Reverse (procurement) auctions

• English descending
• Dutch ascending
• Japanese descending

43
Two yardsticks for good auction design

• Revenue The seller should extract the highest
  possible price
• Efficiency The buyer with the highest valuation
  should get the good
• usually achieved by ensuring incentive
  compatibility bidders are induced to bid their
  true valuation
• maximizing over those bids ensures efficiency.
• The two are sometimes but not always aligned

44
Agents care about utility, not valuation

• Auctions are really lotteries, so you must
  compare expected utility rather than utility.
• Risk attitude speak about the shape of the
  utility function
• linear/concave/convex utility function refers to
  risk-neutrality/risk-aversion/risk-seeking,
  respectively.
• The types of utility functions, and the
  associated risk attitudes of agents, are among
  the most important concepts in Bayesian games,
  and in particular in auctions. Most theoretical
  results about auction are sensitive to the risk
  attitude of the bidders.

45
Connections

• Dutch 1st-price sealed bid
• English Japanese
• English 2nd-price sealed bid under IPV

46
Hints about the analysis of auctions

• Information assumptionsauctions rules
• ( many other assumptions) yield a
• Bayesian game.
• Agents use equilibrium strategies of the Bayesian
  game..
• We now describe some basic results of the theory
  of economic mechanism design in order to show the
  type of studies one can
• carry. We will emphasize the objective of
  revenue optimization.

47
Some Classical Results

• When the agents are risk-neutral, all k-price
  auctions are revenue equivalent (Myerson).
• When agents are strictly risk-averse, then
  
  first-price and Dutch are preferable to
  second and English (Maskin and Riley, Riley and
  Samuelson).

48
Risk-Seeking Agents

• The expected revenue in second-price (English) is
  greater than the expected revenue in first-price
  (Dutch)
• The expected revenue in third-price is greater
  than the expected revenue in second-price
  (English)
• Under constant risk-attitude
• (k1)-price is preferable to k-price

49
Independent Private Value (IPV)versus Common
Value (CV)

• In a CV model agents valuations are correlated.
• the revelation of information during the auction
  plays a significant role
• In the IPV model they are independent.
• Under CV, risk-neutral bidders, we have that
• English gt 2nd gt 1st.

50
The Revenue Equivalence Theorem

• In all auctions for k units with the following
  properties
• Buyers are risk neutral
• IPV, with values independently and identically
  distributed over a,b (technicality
  distribution must be atomless)
• Each bidder demands at most 1 unit
• Auction allocates the units to the bidders with
  the k highest valuations
• The bidder with the lowest valuation has a
  surplus of 0
• a buyer with a given valuation will make the same
  expected payment, and therefore
• all such auctions have the same expected revenue

51
The revelation principle

• In a revelation mechanism agents are asked to
  report their types (e.g.valuations for the good),
  and an action (e.g. decision on the winner and
  his/her payment) will be based the agents
  announcement.
• In general, agents may cheat about their types,
  but
• Any mechanism that implements certain behavior
  (e.g. a good is allocated to the agent with the
  highest valuation,v, and he pays (1-1/n)v) can be
  replaced by (another) revelation mechanism that
  implements the same behavior and where
  truth-revealing is in equilibrium.

52
The revelation principle an example

• In a first-price auction as before, with only two
  participants, submitting half of the valuation is
  in equilibrium.
• Consider the following modification the highest
  bidder wins, but pays half of his bid.
• The new (strange?) protocol implements the same
  function (the same allocation and payments for
  every tuple of agents valuations), and
  truth-revealing is in equilibrium there.

53
Many Participants

• An upper bound -- the expected highest valuation
  (notice that agents may overbid).
• When the number of participants is large --
  English auctions approach the upper bound.
• Marketing is more important than engineering!
  Attract one more participant and you will
  increase your revenue more than in selecting an
  optimized protocol.

54
Multi-Object Auctions

• Several goods
• Bids may be submitted for subsets of goods
• Valuations need not be additive the valuation
  for a set of goods may be different from the sum
  of valuations for the elements it consists of.
• Agents can do better job in expressing their
  valuations

55
Combinatorial bids

• Multiple goods are auctioned simultaneously
• Each bid may claim any combination of goods
• A typical combination a bundle (I bid 100 for
  the TV, VCR and couch)
• More complex combinations are possible

56
Motivation complementarity and substitutability

• Complementary goods have a superadditive
  valuation function
• V(a,b) gt V(a) V(b)
• In the extreme, V(a,b) gtgt0 but V(a) V(b)
  0
• Example different segments of a flight
• Substitutable goods have a subadditive utility
  function
• V(a,b) lt V(a) V(b)
• In the extreme, V(a,b) MAX V(a) , V(b)
• Examples a United ticket and a Delta ticket

57
Multi-Object Auctions the Clarke (GVA)
Mechanism

• Each agent is asked to reveal its valuation for
  each subset of the goods
• An optimal allocation (which maximizes the sum of
  agents valuations, given their reported
  valuations), O, is calculated.
• Agent j is required to pay Aj - Bj, where Aj is
  the sum of other agents (reported) valuations in
  an optimal allocation, Oj, which ignores j, and
  Bj is the sum of other agents (reported)
  valuations in O.
• Truth-revealing is a dominant strategy!
• Notice that in the case of a single good we get
  the second-price (Vickrey) auction.

58
Formal definition of GVA

• Each i reports a valuation function
  possibly different from
• The center calculates which maximizes sum
  of s
• The center calculates which maximizes sum
  of s without i
• Agent i receives (the goods allocated to
  it there) and also a payment of

59
Agent is utility
60
What should agent i bid?

• Of the overall reward
• is bid impacts only
• the auctioneer maximizes
• therefore i should make sure his function is
  identical to the auctioneers!

61
Special case Multiple units of good (example)

• 2 bidders, 3 units (of a single good)
• Bidder As demand curve is (10,8,5), and Bs
  (9,7,6)
• Outcome
• A will win 2 units and B 1 unit
• A will get 9-22-13, i.e. pay 13 for two goods
• B will get 18-23-5, i.e. pay 5 for one good

62
Example (multi-object auction)

• Three goods A, B, C.
• Three agents 1, 2, and 3.
• The valuation assigned by the agents to the
  different goods
• 1 2 3
  
• A 5 3 6
• B 4 4 4
• C 7 4 5
• A,B 8 9 12
• A,C 10 9 10
• B,C 10 12 11
• A,B,C 16 14 14
• AB goes to 3, and C goes to 1, is
  optimal, leading to total of 19.
• Without 1 BC goes to 2, and A
  goes to 3, leading to total of 18.
• Without 2 as in the optimal
  allocation.
• Without 3 BC goes to 2, and A
  goes to 1, leading to a total of 17.
• Final payments 1 pays 18-126, and 3 pays
  17-710 .

63
Other remarks about the Clarke (Generalized
Vickrey Auction) mechanism

• Applies not only to auctions as we know them, but
  to general resources allocation problems
• When externalities exist
• E.g, with public goods
• Not collusion-proof

64
Multi-Object Auctions Maximizing Revenue

• An upper bound -- the expected maximal sum of
  agents valuations over all allocations.
• When the number of participants is large, the
  revenue of the Clarke mechanism approaches the
  upper bound.

65
Competition among sellers

• If there are two sellers that use second-price
  and n agents, then it is likely that about 50 of
  the agents will participate at each auction.
• If one of the sellers deviates to a third-price
  auction, and if an agent prefers second-price to
  third-price then more than 50 may participate
  in the second-price auction.
• However, the expected utility of a buyer is
  identical in all k-price auctions, and therefore
  by deviating to a most profitable auction the
  seller gains but the buyers do not lose!
• Hence, auctions can serve as legal lotteries!

66
Towards implementation computational problems

• Single object auctions are computationally
  tractable.
• Multi-object auctions are in general intractable.
  
• Researchers try and find computationally
  tractable cases, and heuristics are suggested.

67
Computational Aspects Multi-Unit Auctions

• N agents.
• M units of good.
• The value of K units is not greater than the
  value of (K1) units.
• Multi-unit auctions are computationally
• tractable.

68
Computational Aspects Multi-Unit Auctions

• Deciding on optimal allocation given the agents
  bids computationally tractable.
• Applying the Clarke mechanism is tractable.
• Allowing several sets of units of good and
  combinatorial bids on pairs of different goods is
  still tractable.

69
Computational Aspects Network Auctions

• Objects G1,G2,,Gm
• A tree G(V,E) where V is isomorphic to the
  objects
• A bid for a bundle of goods should correspond to
  a path of G
• The case of linear goods (Rothkopf,Nisan) is the
  case where G is a simple path this applicable to
  time scheduling

70
Computational Aspects Network Auctions

• Deciding on optimal allocation (and applying the
  Clarke mechanism) for network auctions is
  computationally tractable
• Other related auction problems can be solved by
  b-matching techniques

71
Motivating Scenario Mechanism design in networks
Jon wishes to sell his TV to one of n potential
buyers. Highest buyers valuation, v , is
known -- Jon can obtain v. The buyers might not
reveal their information in the non-cooperative
setup-- Jon can use a first-price auction.

72
Motivating Scenario mechanism design in networks
Agent 2 might listen to agent 1s message and
submit a slightly higher bid (although his
valuation is much higher)
Mh
1
0
2
M

• Agent 1 can send an encrypted bid.
• This might be quite costly.

.
73
Motivating Scenario Game Theory versus
Cryptography

• A game-theoretic solution
• Use second price instead of first-price --
• The expected revenue in first-price and in
  second-price auctions are identical.
• In second-price auction it is not beneficial
• to listen to others messages.

74
Distributed Games

• Distributed Games
• Game Theory Distributed Systems
• Topology of network
• Syntax of messages
• Asynchronous activity
• Parallelism

75
Implementation in networks a problem

• 2 listens to 1s bid in first price auction

Mh
1
0
2
M
76
Implementation in networks the problem

• Agents messages might be corrupted.
• Classical mechanism design implicitly
• assumes a concrete communication graph.

0
1
0
3
M
2
77
Implementation in networks a solution
Select k

0

6
1
3
Send y
Send k
2
Send k
2
78
Implementation in networks a solution

• Any function that is implementable in the
  classical economic setting is implementable in
  any 2-connected graph.
• The technique relies only on game-theoretic
  assumptions.

79
Parallel Games

• Many locations -- concurrent interactions
• Each user controls several agents (e.g. one at
  each location).
• The interaction at each location is modeled by a
  game in strategic form
• Asynchronous parallel interactions (modeled by a
  probability distribution on the possible
  orderings of interactions).
• Broadcast communication

80
Cooperation in the parallel prisoners dilemma

• Two users. The PD is played at n locations
• (e gt b gt a gt 0)
• (e lt b (b-a)(n-1)/2)
• Equal probability for each ordering of
  interactionsbroadcast communication imply
  cooperation in all locations.
• Distributed systems features yield cooperation in
  finitely repeated PD.

D
C
(b,b)
(-e,e)
C
(e,-e)
(a,a)
D
81
Cooperative outcomes in parallel games

• Assume the same strategic-form game is played in
  all locations
• Assume there is a joint strategy, J, of the
  strategic form game, which is preferable to all
  agents upon the Nash equilibria of it.
• J will be performed in equilibrium of the
  parallel
• game (assuming enough locations).

82
Cooperation Without Enforcement?A comparative
analysis of litigation and online reputation as
quality assurance mechanisms
83
Introduction

• Economic activity requires economic agents to
  abide by the terms of explicit or implicit
  promises.
• Most commercial transactions rely on the legal
  system to assure performance of promises, which
  are written into explicit or implicit contracts.
• The article explores the ability of online
  reputation mechanisms to efficiently induce
  cooperation, compared to contractual arrangements
  depending on the threat of litigation.

84
Introduction cont

• Electronic markets operate on a global scale and
  typically span multiple jurisdictions.
• Litigation across jurisdictions is very costly
  and often infeasible.
• Online reputation mechanisms have emerged as a
  viable alternative to the legal system in such
  settings.
• Information technology is having dramatic impacts
  on the cost, scale and performance of reputation
  mechanisms.

85
Introduction cont

• Online systems greatly reduce the cost of
  collecting and disseminating feedback information
  on a worldwide scale and enable the pooling of
  experiences of unrelated individuals into a
  single, easily accessible repository.
• This increases the likelihood that a feedback
  report for a specific transaction will influence
  large numbers of future transactions, thus
  strengthening the impact of reputation effects.
• Online reputation systems allow the precise
  control of who can participate, what type of
  feedback is solicited, how it is aggregated and
  what type of information is disseminated to the
  community.

86
The setting
87
The setting

• Given is a monopolist seller who in each period
  offers for sale a single unit of a good to m
  buyers.
• Buyer i has valuation wi for a high quality good
  and all buyers value a low quality good at zero.
• Buyer lifetime is exactly one period and in each
  period the m buyers are drawn from the same
  probability distribution, thus buyer valuations
  are independent and identically distributed
  within and across periods.
• There is an infinite number of periods and the
  seller has a period discount factor ? reflecting
  the frequency of transactions within the
  community, or the probability that the game will
  end after each period.

88
The setting cont

• Seller effort determines the probability that the
  good provided will be of low quality if the
  seller exerts low effort, the good will be of low
  quality with probability ?, whereas if the seller
  exerts high effort he will incure additional cost
  c and the good will be of low quality with a
  smaller probability ? (?lt?).
• The sellers objective is to maximize the present
  value of his payoffs over the entire span of the
  game, while the buyers objective is to maximize
  their short-term (stage game) payoff.

89
The setting cont

• In each period a mechanism is used to allocate
  the good among the m buyers by determining the
  buyer that receives the good and the price she
  pays to the seller.
• Assume a second price Vickrey auction is used to
  award the good to the highest valuation buyer who
  pays a price equal to the second-highest bid G.

90
The reputation mechanism model
91
The reputation mechanism model

• The reputation mechanism allows buyers to rate
  the seller based on the quality of the good
  received.
• Buyers report the outcome of a transaction as
  either positive or negative.
• positive rating indicate high quality good
  received.
• negative rating indicate low quality good
  received.
• The mechanism aggregates past ratings and
  publishes a summary of the sellers most recent
  ratings.
• Buyers can see the total number of each type of
  rating received by the seller during the most
  recent N transactions (earlier ratings are
  discarded).

92
The reputation mechanism model cont

• A sellers feedback profile is represented as (x,
  N), where x?0,1,,N is the number of negative
  ratings currently contained within that window.
• At the end of each period, the ratings received
  during the current period is added to the profile
  whereas the rating received N periods ago is
  discarded.

93
The role of information technology in online
reputation systems

• Once an online system has been developed, the per
  period cost of collecting, processing, and
  communicating ratings information is much lower
  compared to a traditional off-line system.
• The type of structured design for the reputation
  mechanism in the suggested setting is only
  feasible in the context of an online system.

94
The role of information technology in online
reputation systems cont

• Since the cost of providing feedback is low
  enough, customers are given incentives that
  induce participation and truth-telling.
• Information technology makes the outcome of any
  single transaction immediately known to the
  entire population of prospective buyers,
  therefore it increases the proportion of
  transactions affected by the sellers reputation.

95

• The above affects significantly the ability of
  the reputation mechanism to promote cooperative
  behavior.
• Lets focus on the special case where N1. This
  corresponds to a reputation mechanism that
  publishes the single most recent rating received
  for the seller.
• A sellers reputation is denoted by a binary
  state variable
• x ?0,1.

96
Stage game for reputation mechanism

• Seller offers a single unit of a good, promising
  to deliver a high quality good.
• System provides a binary (positive or negative)
  rating for the seller, based on the buyer in the
  most recent period.
• Buyers bid their expected valuations for the good
  in second price Vickrey auction. The winning
  bidder pays G, which is the second-highest bid.
• Denote by w1 and w2 the respective valuations
  for a high quality good of the winning bidder and
  the second-highest bidder.

97
Stage game for reputation mechanism cont

• Seller decides whether to exert high effort at
  cost c, or low effort at cost 0, with
  corresponding probabilities that the resulting
  good is of low quality being ? and ? (?lt?).
• Buyer receives the good, experiences its quality,
  and realizes the corresponding valuation w1 for a
  high quality of good or 0 for low quality good.
  Buyer reports the quality of the good received to
  the system, and the rating of the seller reported
  in the next period is changed accordingly.

98
Characterization of Equilibrium Outcomes
99
Characterization of Equilibrium Outcomes

• Let s(x,ht) ? 0,1 denote the sellers strategy
  in period t, equal to the probability the seller
  will cooperate in period t if his current
  reputation profile contains negative ratings at
  the beginning of the period and the past history
  of play is ht.
• Restrict the seller to stationary strategies,
  where s(x,ht) does not depend on t, or the
  history of play.
• Let ss(0), s(1) denote the sellers strategy
  vector.

100
Characterization of Equilibrium Outcomes cont

• Expected auction revenue
• Expected surplus for winning bidder
• Sellers payoff

101
Characterization of Equilibrium Outcomes cont

• Let U(x,s) denote the sellers expected future
  payoff.
• Since the reputation mechanism discards past
  ratings, the sellers future payoff is
  independent of the current state x of his
  reputation profile.
• Therefore

102
Characterization of Equilibrium Outcomes cont

• A strategy s is an equilibrium strategy if and
  only if it satisfies the incentive compatibility
  constrains
• We will focus the attention on the Pareto
  dominant equilibrium strategy s that maximizes
  the sellers expected discounted lifetime payoff
• Where x0 ? 0,1 is the initial state of the
  reputation profile of new sellers, as well as the
  winning buyers expected single period surplus.

103
Proposition 1

• Proposition 1 summarizes the sellers optimal
  strategy.
• Let .

104
Following proposition 1

• The condition
• in order to induce any degree of cooperation, the
  buyers valuation of high quality must be high
  enough relative to the incremental cost of
  exerting high effort, so that discounted future
  payoffs from sustained cooperation are greater
  than short-term gains from cheating.

105
Following proposition 1

• The condition
• in order to induce any degree of cooperation, the
  sellers must transact with sufficient frequency
  within the span of the reputation mechanism so
  that the stem of future payoffs is large enough
  to offset the short- term gains from cheating.

106
Total surplus
107
Total surplus

• Single stage total surplus
• Average single stage total surplus

108
Proposition 2

• Proposition 2 shows the total surplus
  corresponding to the seller strategy of
  proposition 1

109
Enforcement-based Framework (litigation)

• Instead of reporting the quality of good
  received, the buyer may sue the seller for
  failing to deliver a high quality good.

110
Stage game for litigation mechanism

• Seller offers a single unit of a good, promising
  to deliver a high quality good.
• Buyers bid their expected valuations for the good
  in second price Vickrey auction. The winning
  bidder pays G, which is the second-highest bid.
• Denote by w1 and w2 the respective valuations
  for a high quality good of the winning bidder and
  the second-highest bidder.
• Seller decides whether to exert high effort at
  cost c, or low effort at cost 0, with
  corresponding probabilities that the resulting
  good is of low quality being ? and ? (?lt?).

111
Stage game for reputation mechanism cont

• Buyer receives the good, experiences its quality,
  and realizes the corresponding valuation w1 for a
  high quality of good or 0 for low quality good.
• Buyer decides whether or not to sue seller. If
  buyer does not sue, the stage game ends.
• If the buyer sues, the court finds for the buyer
  with probability a if the good received was high
  quality, and with higher probability b if the
  good received was low quality. Independent of the
  court decision, each party incurs litigation
  costs L.
• If court finds for the buyer, then the seller has
  to pay to the buyer damages D.

112
Stage game for litigation mechanism
113
Proposition 3

• Each period is independent, therefore, analysis
  of the game consists of analyzing the stage game.
• Prop 3 shows the resulting outcomes of the
  litigation game

114
Proposition 4

• Implication of proposition 3 for maximizing total
  surplus

115
Following proposition 4

• If the tree conditions
• are simultaneously satisfied, then the buyer
  will sue only when a low quality good is
  received the seller will always exert high
  effort because its cost is less than the expected
  reduction in legal costs and damages and
  inducing the seller to exert high effort through
  the threat of litigation increases the total
  surplus.

116
Discussion
117
Impact of information technology

• Before the advent of the internet, word-of-mouth
  regarding professionals and merchants took place
  within relatively small and (almost) mutually
  disjoint groups of neighbors, friends,
  co-workers, etc.
• The above is equivalent to a setting where each
  group operates an independent reputation
  mechanism that only receives and disseminates
  feedback from members of the group.
• If a seller operates over a large, fragmented
  territory, the number of such groups would be
  large.

118
Impact of information technology cont

• The sellers discount factor for future payoffs
  from any given group would be smaller
• If the seller discounts the future by ? per
  period and transacts with each group every n-th
  period on average, the appropriate discount
  factor in considering the sellers behavior for
  each group will be .
• Since ? lt 1, for large enough n it will be
  .
• Therefore, reputation mechanisms will fail to
  induce cooperation when feedback networks are
  sufficiently fragmented.

119
Impact of information technology cont

• Internet-based online reputation mechanism
  provide easily accessible, low cost focal points
  for previously disjoint groups to pool their
  experiences with service providers and merchants
  into a single feedback repository.
• As these feedback mechanisms cover more groups,
  the effect is equivalent to reducing the degree
  of fragmentation n of the feedback networks. This
  increases the discount factor of the seller.
• At the limit, outcome of any single transaction
  becomes immediately known to the entire
  population of prospective buyers. This would
  result in the sellers discount factor getting
  closer to 1.

120
Impact of information technology cont

• As ? increases, reputation becomes more efficient
  relative to litigation.
• Reputation may be less efficient than regulation
  for lower values of ?.
• Reputation may become more efficient than
  litigation as ? increases and approaches 1.

121
Comparing the efficiency of reputation vs.
litigation mechanism

• The reputation mechanism induces the seller to
  exert high effort most of the time provided that
  .
• A seller with good reputation will always
  cooperate, while a seller with bad reputation
  will cooperate with probability
  .
• Average total surplus per period

122
Comparing the efficiency of reputation vs.
litigation mechanism cont

• The reputation mechanism reduces the total
  surplus by
• compared to the high-effort first-best outcome.
• The efficiency of litigation mechanism depend on
  the litigation costs L.
• If then for a properly
  selected level of damages D the litigation
  mechanism will induce the seller to always
  cooperate.

123
Comparing the efficiency of reputation vs.
litigation mechanism cont

• The surplus, in this case, is
• the reputation mechanism reduces total surplus by
  2?L compared to high effort first-best outcome.
• If ??1 and if w1?w2 then the reduction in surplus
  for the reputation mechanism simplifies to
  .
• In this case, the reputation mechanism is more
  efficient than litigation in terms of total
  surplus generated if and only if

124
Comparing the efficiency of reputation vs.
litigation mechanism cont

• The crucial determinant of the relative
  efficiency of the two mechanisms is the magnitude
  of litigation costs L relative to the incremental
  cost of high effort c. The higher the ratio, the
  more attractive the use of reputation mechanism
  relative to litigation.
• For values of ? and ?,
  the two mechanisms are comparable in inducing
  cooperation when litigation costs are between
  50-100 of the incremental cost of high effort.
  If litigation costs rise above that threshold,
  reputation emerges as most efficient mechanism.

125
Conclusions

• The effectiveness of a reputation mechanism in
  inducing cooperative behavior has a discontinuous
  relationship to the frequency of transactions
  that are affected by this mechanism A certain
  degree of participation is required before
  reputation can induce a significant level of
  cooperation.
• If legal costs are comparable to or larger than
  the incremental cost of cooperation, reputation
  mechanisms are likely to outperform litigation in
  terms of inducing cooperation and maximizing
  total surplus.