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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BATTLE OF SEXES

Husband

Wife

Basket-ball

Opera

(3,1)

Basketball

(0,0)Opera

(0,0)(1,3)

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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)