The Core
A Methodological Toolkit to Reform Payment Systems
Game Theory
World Bank, Washington DC, November 5th, 2003
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¶ The need for Game Theory in payment systems reforms.
¶ Basic concepts.
¶ Applications to payment systems.
¶ Conclusions.
CONTENTS
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GAME THEORY AND PAYMENT SYSTEMS REFORMS
• Game theory analyzes the interaction between several agents (players) to understand what strategy (or set of strategies) each player chooses and what is the outcome generated
• Plenty of interactions and consequent conflicts arise over payment systems, especially when a reform has to be decided and implemented
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CONFLICTS IN PAYMENT SYSTEMS
• Conflicts between Large and Small Banks over the access of small banks to systematically important payment systems
• Conflicts between dealers/brokers and banks and other non bank financial institutions over the access of the former to large value systems and securities settlement systems
• Banks and other non bank financial institutions over the access to ACH and large value systems
Example of potential conflictIssue
Access
• Conflicts between Public Authorities: Central Bank vs. Competition Authority over
respective roles in competition issues Central Bank vs. Legislative Authority over
legislative role Central Bank as the payment system overseer and
other regulators (e.g. Securities Commission and Committee Supervisors)
Power
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CONFLICTS IN PAYMENT SYSTEMS (cont.)
• Vertical conflicts between all the stakeholders, over the technology to be used
• Horizontal conflicts between:Different service providers (different incumbents, incumbent vs. entrants)Different participants (different incumbents, incumbent vs. entrants)
Example of potential conflictIssue
Service provision
• Vertical conflicts between all the stakeholders, over the price to be used
• Horizontal conflicts between:Different service providers (different incumbents,
incumbent vs. entrants)Different participants (different incumbents,
incumbent vs. entrants)
Pricing
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CONFLICTS IN PAYMENT SYSTEMS (cont.)Example of potential conflictIssue
• Conflicts between system providers and system participants over the participants’ compliance of the rules of the payment system
• Conflicts between regulators and system providers over the system’s ability to deliver efficiency and safety
Compliance
• Conflicts between national and international public institutions and private sector over who should fund a payment system reform and to what conditions
Funding
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¶ The need for Game Theory in payment systems reforms.
¶ Basic concepts.
¶ Applications to payment systems.
¶ Conclusions.
CONTENTS
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• Gibbons, R., “Game Theory for Applied Economists”, Princeton University Press, 1992
• Osborne, M. and Rubinstein, A., “A Course in Game Theory, MIT Press, 1994
BIBLIOGRAPHY
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KEY ELEMENTS OF A GAME
• A player’s payoffs must depend on other players’ payoffs
Strategies
• There must be at least two players to have a game
• There must be at least two strategies available for each player to have a game
Payoffs
Players
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All Games
TYPES OF GAME
Incomplete Information
Dynamic
Complete Information
Static
Dynamic
Static
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NASH EQUILIBRIUM
• To solve a static game with complete information Nash Equilibrium (NE) is the equilibrium concept
• It is defined as a set of strategies, one for each player, so that, given the other players’ strategies, no player has any incentive to change her own strategy
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PRISONER’S DILEMMA
Prisoner 2
Priso-ner 1
Confess
No
(-1,-1)
Confess
(-5,0)No
(0,-5)(-3,-3)
•The pair of strategies of P1 and P2 (Confess, Confess) is the NE of the game
•Both players, given the other player’s strategy, always prefer Confess
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FEATURES OF EQUILIBRIA
•In the Battle of Sexes two pure strategy NE: (B,B) and (O,O)
Husband
Wife
B
O
(3,1)
B
(0,0)O
(0,0)(1,3)
1. Multiplicity of equilibria may occur
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FEATURES OF EQUILIBRIA (cont.)
•In the Prisoner’s Dilemma the unique NE (Confess, Confess) is a sub-optimal outcome
P.2
P.1
Confess
No
(-1,-1)
Confess
(-5,0)No
(0,-5)(-3,-3)
2. NE outcomes might not be Pareto efficient
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FEATURES OF EQUILIBRIA (cont.)
•In the Matching Pennies game on the right, there is no NE in pure strategies
P2
P1
H
T
(-1,1)
H
(1,-1)T
(1,-1)(-1,1)
3. There might not be any NE in pure strategies
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All Games
TYPES OF GAME
Incomplete Information
Dynamic
Complete Information
Static
Dynamic
Static
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SUBGAME PERFECT EQUILIBRIUM
• To solve a dynamic game with complete information Subgame Perfect Equilibrium (SPE) is the equilibrium concept
• It is defined as a set of strategies for each player so that, for all subgames (given all history paths), the strategy profile is a NE of the game
• It has to be solved (if finite) through backward induction (starting from the terminal nodes of the game)
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EXAMPLE
P2
P1
(2,1)
•The only SPE is (L,r)
•Notice: the pair (R,l) is not a SPE because P2 will always play r once it should become its turn to play
RL
l r
(0,0)
(1,2)
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REPEATED GAMES AND COOPERATION
• If the prisoner’s dilemma above described is repeated several times there is room for a SPE where players cooperate
• The following strategy is a SPE of the prisoner’s dilemma repeated for several periods and for sufficiently high values of discount rates for both players:Play No as long as the
other player does soPlay Confess forever on
after the stage when the other player plays confess
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¶ The need for Game Theory in payment systems reforms.
¶ Basic concepts.
¶ Applications to payment systems.
• Access game: small banks in S.I.P.S.
• Pricing game (vertical).
¶ Conclusions.
CONTENTS
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• A National Payment Council (NPC) must decide de facto – through pricing and level of sophistication of the system – whether small banks should access directly systematically important payment system (SIPS) or through the intermediation of large banks
• The assessment is taken through voting: majority wins
• Small banks and large banks are both represented by their own association in the NPC. The leader of the NPC (and other voter) is the Central Bank.
CONTEXT
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PLAYERS AND STRATEGIES
Strategies
• Each player can either support a system design which favors the access of small banks to SIPS (In) or be against it (Out) by voting simultaneously
• Central Bank (CB)
• Large Banks (L)
• Small Banks (S)
Players
Static game
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OUTCOMES
CB: Out
S
LIn
OutIn
Out OUT
CB: In
S
LIn
Out
OUT
In
Out
ININ (price
war)
IN (price
war)OUT
OUTIN
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PAYOFFS
CB: Out
(0,0,0)
CB: In
(0,0,0)
(0,0,0)(-3,3,4)
S
LIn
OutIn
Out(0,0,0)(-5,1,3)
(-5,1,3)(-3,3,5)
S
LIn
OutIn
Out
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• There are two pure strategy NE: (In, In, In) and (Out, Out, Out)
• The Central Bank can cause the NE (In, In, In), by:
Changing the voting systems
Committing in advance to the vote In
Pushing on large banks. It could state for example that, even if the outcome of the vote is Out, it would either create a huge pressure on them to reduce their fees, or it would oblige them to set very low fees to end-users.
NASH EQUILIBRIA
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¶ The need for Game Theory in payment systems reforms.
¶ Basic concepts.
¶ Applications to payment systems.
• Access game: small banks in S.I.P.S.
• Pricing game (vertical).
¶ Conclusions.
CONTENTS
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• A fee system, composed of fees to the participants and fees to the end-users has to be designed within the National Payment Council (NPC), and the various stakeholders of the NPC must agree upon it
• The Central Bank is also the system provider
CONTEXT
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PLAYERS AND STRATEGIES
Strategies
• CB: set fees to participants
• P: set fees to end-users
• E: set the number of transactions
Players
• Central Bank (CB)
• Participants (P)
• End-users (E)Static game
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PAYOFFS
E
Variables affecting payoffs
P
• Payoffs of system provider
• Payoffs of participants
• Payoffs of end-users
• Number of transactions
• Fees to end-users
• Number of transactions
• Fees to participants (FP)
• Fees to end-users (FE)
CB
Player
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NASH EQUILIBRIUM
FP
FE
FP(FE)
NE of the game FE(FP)
• In FP(FE) on the figure CB sets its own optimal FP for any given FE set by P
• Analogously in FP(FE) P set their own optimal FE
for any given FP set by CB
• Point A is the NE of the game
• It is possible to improve upon A by imposing costs / subsidizing participants / end-users
A
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¶ The need for Game Theory in payment systems reforms.
¶ Basic concepts.
¶ Applications to payment systems.
¶ Conclusions.
CONTENTS
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FINAL REMARKS
• Game theory can be definitely a useful tool for payments systems reforms: it could be adopted initially within the National Payment Council
• It requires two conditions to be respected (too cumbersome game otherwise)
Honesty by the players (as much as possible)
Simple assumptions (80:20 approach)
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