Airlines Model

A Model On

DOMESTIC AIRLINE INDUSTRY IN INDIA

Using Graph Theory and Game Theory

©Rupakshi Bhatia

FOCAL AIRLINE:

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The Indigo Destinations Map

[8].

©Rupakshi Bhatia

THE PROBLEMS

1. What are the least number of “hops” that one would have to make to travel from one city to another?

2. How do we choose the flight that suits us the best?

3. What determines the pricing of Airline tickets?

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WHY GRAPH THEORY AND GAME THEORY?

• Graph Theory allows us to pictorially represent the relationship between airline routes and cities, and thus helps us to deal with Problems 1 and 2.

• Game Theory allows us to study the behavior of businessmen handling firms and their reasons behind the pricing of airline tickets, thus, helping us to address Problem 3.

©Rupakshi Bhatia

THE GRAPH

The graph is formally given in the form G = (V(G), E(G), ∑, lab)where

• V(G) is a finite set of vertices; |V(G)|=28 • E(G) ⊆ V2 is a set of undirected edges• ∑ is a finite nonempty set of vertex labels, and • lab: V(G)∑ is a labelling function where for v ϵ V(G), lab(v) is the label of vertex v [1].

Remarks:1. Graph G is not a Complete Graph.2. Graph G is a Connected Graph.

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v lab(v)

Agartala v1

Ahmedabad v2

Bangalore v3

Bhubaneswar v4

Chandigarh v5

Chennai v6

Coimbatore v7

Delhi v8

Dibrugarh v9

Goa v10

Guwahati v11

Hyderabad v12

Imphal v13

Indore v14

Jammu v16

Kochi v17

Kolkata v18

Lucknow v19

Mumbai v20

Nagpur v21

Patna v22

Pune v23

Raipur v24

Srinagar v25

Trivandrum v26

Vadodara v27

Vishakhapatnam v28

v lab(v)

Jaipur v15

©Rupakshi Bhatia

The Adjacency Matrix

• 28 x 28 symmetric binary matrix A = [aij] such that:

aij = 1, if {vi, vj} ϵ E(G);= 0, otherwise

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The Adjacency Matrixaga ahm ben bhu cha che coi del dib goa guw hyd imp ind jai jam koc kol luc mum nag pat pun rai sri tri vad vis

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 v24 v25 v26 v27 v28

aga v1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

ahm v2 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0

ben v3 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 0 1 1 1 1 0 1 0 0 1 0 0

bhu v4 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1

cha v5 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

che v6 0 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1

coi v7 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0

del v8 0 1 1 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 0 1 0

dib v9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

goa v10 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0

guw v11 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

hyd v12 0 1 1 1 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 1 1 0 1 1 0 0 0 1

imp v13 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

ind v14 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0

jai v15 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0

jam v16 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

koc v17 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0

kol v18 1 1 1 1 0 1 0 1 1 0 1 1 1 0 1 0 0 0 0 1 1 1 0 1 0 0 0 1

luc v19 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0

mum v20 0 1 1 1 1 1 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 0 0 0 1 1 1 0

nag v21 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0

pat v22 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0

pun v23 0 1 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

rai v24 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0

sri v25 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0

tri v26 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0

vad v27 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

vis v28 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

©Rupakshi Bhatia

• A Flight(F) is an open path between any two vertices (cities) of the graph.

It is a finite sequence of the form vi0, ej1, vi1, ej2, …, ejk, vik, which consists of alternating vertices and edges of G, where vix ≠ viy ∀ x≠y and ejx ≠ ejy ∀ x≠y.

vi0 is the initial city (vertex) and vik is the terminal city (vertex).

k is the length of the flight. A flight cannot be a zero-length path.

• Defining, two cities vx and vy are CONNECTED if there is a DIRECT FLIGHT between them.

©Rupakshi Bhatia

• 28 x 28 symmetric matrix C = [cij] such that:

cij = kij, if {vi, vj} ϵ E(G);= 0, otherwise

Here, kij is the number of edges joining vi and vj.

The Connection Matrix

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The Connection Matrixaga ahm ben bhu cha che coi del dib goa guw hyd imp ind jai jam koc kol luc mum nag pat pun rai sri tri vad vis

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 v24 v25 v26 v27 v28

aga v1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

ahm v2 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0

ben v3 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 0 1 1 1 1 0 1 0 0 1 0 0

bhu v4 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1

cha v5 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

che v6 0 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1

coi v7 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0

del v8 0 1 1 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 0 1 0

dib v9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

goa v10 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0

guw v11 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

hyd v12 0 1 1 1 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 1 1 0 1 1 0 0 0 1

imp v13 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

ind v14 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0

jai v15 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0

jam v16 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

koc v17 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0

kol v18 1 1 1 1 0 1 0 1 1 0 1 1 1 0 1 0 0 0 0 1 1 1 0 1 0 0 0 1

luc v19 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0

mum v20 0 1 1 1 1 1 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 0 0 0 1 1 1 0

nag v21 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0

pat v22 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0

pun v23 0 1 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

rai v24 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0

sri v25 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0

tri v26 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0

vad v27 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

vis v28 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

©Rupakshi Bhatia

• The Connection Matrix of G is the same as the Adjacency Matrix of G since the maximum number of edges between any two vertices of G is 1.

• Out of a total of 28C2=378 pairs of vertices, about 6 cases existed where vx was connected to vy but vy was not connected to vx (vx, vy ϵ V(G) ) . Since 6/378=0.015873 is a very small probability, we ignore such possibilities and assume that CONNECTED(vx, vy)=T ⇔ CONNECTED (vy, vx)=T.

Thus,• If there is a Direct Flight from vx to vy, then there is a Direct Flight from vy to vx as well.• The Adjacency and Connection Matrices are symmetric.

©Rupakshi Bhatia

The Degree Matrix of G

• 28 x 28 diagonal matrix D = [dij] such that:

dij = deg(vi), if i=j;= 0, if i ≠ j.

• Degree of a vertex v, deg(v) is the number of edges incident with the vertex v, with loops counted twice.

©Rupakshi Bhatia

The Degree Matrix of Gaga ahm ben bhu cha che coi del dib goa guw hyd imp ind jai jam koc kol luc mum nag pat pun rai sri tri vad vis

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 v24 v25 v26 v27 v28

aga v1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ahm v2 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ben v3 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

bhu v4 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

cha v5 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

che v6 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

coi v7 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

del v8 0 0 0 0 0 0 0 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dib v9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

goa v10 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

guw v11 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

hyd v12 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

imp v13 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ind v14 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0

jai v15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0

jam v16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0

koc v17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0

kol v18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0

luc v19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0

mum v20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0

nag v21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0

pat v22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0

pun v23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0

rai v24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0

sri v25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0

tri v26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0

vad v27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0

vis v28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4

©Rupakshi Bhatia

• 28 x 28 matrix P(G) = [pij] such that:

pij = 0, if i=j;= x otherwise.

Here, x is the length of the shortest path between vi and vj.

The Path Matrix of G

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The Path Matrix of Gaga ahm ben bhu cha che coi del dib goa guw hyd imp ind jai jam koc kol luc mum nag pat pun rai sri tri vad vis

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 v24 v25 v26 v27 v28

aga v1 0 2 2 2 3 2 3 2 2 3 1 2 1 3 2 3 3 1 3 2 2 2 3 2 3 3 3 2

ahm v2 2 0 1 2 2 1 2 1 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 2 2 2

ben v3 2 1 0 2 1 1 1 1 2 1 1 1 2 2 1 2 2 1 1 1 1 2 1 2 2 1 2 2

bhu v4 2 2 2 0 2 2 2 1 2 2 2 1 2 2 2 2 2 1 2 1 2 2 2 2 2 2 2 1

cha v5 3 2 1 2 0 2 2 2 3 2 2 2 3 2 2 3 2 2 2 1 2 3 2 3 2 2 2 3

che v6 2 1 1 2 2 0 1 1 2 2 2 1 2 2 2 2 2 1 2 1 2 2 1 2 2 1 2 1

coi v7 3 2 1 2 2 1 0 1 3 2 2 1 3 2 2 2 2 2 2 1 2 2 2 2 1 2 2 2

del v8 2 1 1 1 2 1 1 0 2 1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 1 2 1 2

dib v9 2 2 2 2 3 2 3 2 0 3 2 2 2 3 2 3 3 1 3 2 2 2 3 2 3 3 3 2

goa v10 3 1 1 2 2 2 2 1 3 0 2 2 3 1 2 2 2 2 2 1 2 2 2 2 2 2 2 3

guw v11 1 2 1 2 2 2 2 1 2 2 0 2 1 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2

hyd v12 2 1 1 1 2 1 1 1 2 2 2 0 2 2 1 2 1 1 1 1 1 2 1 1 2 2 2 1

imp v13 1 2 2 2 3 2 3 2 2 3 1 2 0 3 2 3 3 1 3 2 2 2 3 2 3 3 3 2

ind v14 3 2 2 2 2 2 2 1 3 1 2 2 3 0 2 2 2 2 2 1 1 2 2 1 2 2 2 3

jai v15 2 1 1 2 2 2 2 2 2 2 2 1 2 2 0 3 2 1 2 1 2 2 2 2 2 2 2 2

jam v16 3 2 2 2 3 2 2 1 3 2 2 2 3 2 3 0 2 2 2 2 2 2 2 2 2 3 2 3

koc v17 3 2 2 2 2 2 2 1 3 2 2 1 3 2 2 2 0 2 1 1 2 2 2 2 2 1 2 2

kol v18 1 1 1 1 2 1 2 1 1 2 1 1 1 2 1 2 2 0 2 1 1 1 2 1 2 2 2 1

luc v19 3 2 1 2 2 2 2 1 3 2 2 1 3 2 2 2 1 2 0 1 2 1 2 2 2 2 2 2

mum v20 2 1 1 1 1 1 1 1 2 1 2 1 2 1 1 2 1 1 1 0 1 2 2 2 1 1 1 2

nag v21 2 2 1 2 2 2 2 1 2 2 2 1 2 1 2 2 2 1 2 1 0 2 1 2 2 2 2 2

pat v22 2 2 2 2 3 2 2 1 2 2 2 2 2 2 2 2 2 1 1 2 2 0 2 2 2 3 2 2

pun v23 3 1 1 2 2 1 2 1 3 2 2 1 3 2 2 2 2 2 2 2 1 2 0 2 2 2 2 2

rai v24 2 2 2 2 3 2 2 1 2 2 2 1 2 1 2 2 2 1 2 2 2 2 2 0 2 3 2 2

sri v25 3 2 2 2 2 2 1 1 3 2 2 2 3 2 2 2 2 2 2 1 2 2 2 2 0 3 2 3

tri v26 3 2 1 2 2 1 2 2 3 2 2 2 3 2 2 3 1 2 2 1 2 3 2 3 3 0 2 2

vad v27 3 2 2 2 2 2 2 1 3 2 2 2 3 2 2 2 2 2 2 1 2 2 2 2 2 2 0 3

vis v28 2 2 2 1 3 1 2 2 2 3 2 1 2 3 2 3 2 1 2 2 2 2 2 2 3 2 3 0

©Rupakshi Bhatia

• Service Frequency(f) :f: V x V ℕf(vx, vy) := Number of Flights from vx to vy per day

Assume that: • f(vx, vy) is constant throughout the week.• Aircrafts are large enough to satisfy expected demands; if not, frequency is increased to meet excess demands. • People value (i) Time(ii) Money(iii) Comfort

• Busiest Airport(s) or HUB(S) : Airport vertex with maximum degree.As can be seen from the degree matrix, the Busiest Airport is that of Delhi with degree 20.

©Rupakshi Bhatia

Value of a Flight

Utility Function:X= {Time, Money, Comfort} is a set of parameters.U: X ℝ+ is a consumer-specific Utility Function representing the Preference Relation R ϶aRb ⇒ a » b and∀ a, b ϵ X , a » b ⇔ U(a) ≥ U(b)

whereR ⊂ X x X [5].

• a » a is true ∀ a ϵ X ⇒ R is Reflexive.• a » b ⇒ b » a. ∴ R is not Symmetric.• ∀a, b ∈ X, a » b ∧ b » a ⇒ a = b is not true. ∴ R is not Antisymmetric [4].• a » b ∧ b » c ⇒ a » c is true. ∴ R is Transitive.

©Rupakshi Bhatia

Then, Value of a Flight, Val(F) = U(Time).Q(Time) + U(Money).Q(Money) + U(Comfort).Q(Comfort)whereQ(Time)=Quantity of Time savedQ(Money)=Quantity of Money savedQ(Comfort)=Quantity of Extra Comfort received.

Then, a customer will choose the Flight with the greatest Value.

For example, suppose•‘n’ choices of Flights {F1, F2, … Fn}• Take {t1, t2, …, tn} hours from the initial city to the customer’s destination,• Price {P1, P2, …, Pn} • Comfort level {C1, C2, …, Cn}.

Let t = max(t1, t2, …, tn) , P = max(P1, P2, …, Pn) , C=min(C1, C2, …, Cn).Then, Val(Fi)= [ U(Time).(t-ti) + U(Money).(P-Pi) + U(Comfort).(Ci-C) ]

©Rupakshi Bhatia

If a journey involves taking multiple flights, Value of a journey = (Sum of values of all flights involved/Number of flights involved)-[U(Time). (Waiting time) ]

-c

The journey with the greatest value is then chosen.

©Rupakshi Bhatia

DOMESTIC AIRLINE INDUSTRY: AN OLIGOPOLY

What is an oligopoly?It is a market structure in which only few sellers, each having a large market share offer similar or identical products [2].

‘Highly concentrated Markets’: Where most of the total market share is locked up by a small number of firms [3].Thus, an oligopoly is a highly concentrated market.

• One commonly used method of market concentration is the Herfindahl-Hirschman Index (HHI), given by

HHI = ∑Si2

i=1

N

whereSi denotes the percent of market controlled by the ith firmN denotes the number of firms in the market

©Rupakshi Bhatia

Then,HHI > 1,800 denotes a highly concentrated market [6].

We have

[7].

*As in July 2012©Rupakshi Bhatia

Thus, HHI for the Domestic Airline Industry in India = (20.7)2 + (18.3)2 + (6.9)2 + (27.3)2 + (19.5)2 + (7.4)2

= 1,991.29 > 1,800.

Therefore, Airline Industry is an Oligopoly.

©Rupakshi Bhatia

PRICING STRATEGIESPrice Discrimination

Price Discrimination: The practice of selling the same good/service at different prices to different customers [2].Here, the service is a Flight ticket from vx to vy.

Indigo practices Price Discrimination in 3 ways:

1. One Way Ticket/ Round Trip TicketsPrice(Round Trip) < Price(One Way).

2. Age0, if 0 ≤ Age < 2

Let Ai= 1, if 2 ≤ Age < 122, if Age ≥ 12

for the ith customer.Then, A1 < A2 ⇒ Price(A1) < Price(A2).

{ ©Rupakshi Bhatia

3. TimePrice(t) is an increasing function wheret ϵ [-365, 0)

Then,t1 < t2 ⇒ Price(t1) ≤ Price(t2) [8].

©Rupakshi Bhatia

Does Indigo have an incentive to change price unilaterally?

• All other factors kept constant.• Set of Player, S = {A, B} where Player A: Indigo, Player B: Combined others• Strategy Set for A, SA = {Raise, Lower, Do Nothing}• Strategy Set for B, SB = {React, Don’t React}• Single-shot game• Game is not symmetric since SA ≠ SB

• Pay-offs represent monetary gains.

React Don’t React

Raise (20, 20) (-10, 40)

Lower (-5, -5) (80, -100)

Do Nothing (0, 0) (0, 0)

A

B

©Rupakshi Bhatia

• What A would want: If A raises its price, B should reactIf A lowers its price, B should not react.

• What would actually happen(since the market is an Oligopoly): If A raises its price, B does not reactIf A lowers its price, B reacts.

Why?

•∏B(Raise, Don’t React)=40 ≥ ∏B(Raise, React)=20∴∏B(Raise, Don’t React) ≥ ∏B(Raise, sB) ∀ sB ϵ SB

• ∏B(Lower, React)=(-5) ≥ ∏B(Lower, Don’t React)=(-100)∴∏B(Lower, React) ≥ ∏B(Lower, sB) ∀ sB ϵ SB

Where ∏j(s, s’) is the payoff for player j when PA plays strategy s and PB

plays strategy s’ in a 2-player game.

©Rupakshi Bhatia

Thus, The pay-off matrix for player A reduces to:

Raise -10

Lower -5

Do Nothing 0

Clearly,∏A(Do Nothing)=0 ≥ ∏A(s); s ϵ SB \ {Do Nothing}

∴ There is no incentive for A to change its price unilaterally, other factors remaining constant. So, a change in price is due to some other factors/conditions only.

A

©Rupakshi Bhatia

Lowering and Raising of prices

Now, suppose A is incurring heavy monetary losses due to erosion of customer base.∴ A tries to reduce its losses by increasing its market share.Then,

React Don’t React

Raise 20 -10

Lower -5 80

Do Nothing -50 -50

A

B

©Rupakshi Bhatia

Given that:B reacts when A lowers priceB does not react when A raises price.

∴ Pay-off matrix of A reduces to

Raise -10

Lower -5

Do Nothing -50

Clearly,∏A(Lower)=(-5) ≥ ∏A(s); s ϵ SB \ {Lower}

Thus, A lowers its price B reacts by lowering Price War Airline prices crash.

A

©Rupakshi Bhatia

• Prices crash till they reach the ‘Crash Point’ where Revenue < Cost of production(service).

• Now, the game changes with changing pure strategy sets;SA = SB = {Compete, Collude}

Compete Collude

Compete (-5,-5) (0,0)

Collude (0,0) (20,20)

Claim: (Collude, Collude) is the Nash Equilibrium.

A

B

©Rupakshi Bhatia

Proof:•∏A(Collude, Collude)=20 ≥ ∏A(Compete, Collude)=0∴∏A(Collude, Collude) ≥ ∏A(sA , Collude) ∀ sA ϵ SA

• ∏B(Collude, Collude)=20 ≥ ∏B(Collude, Compete)=0∴∏B(Collude, Collude) ≥ ∏B(Collude, sB) ∀ sB ϵ SB

• Thus, firms collude prices rise

• Prices rise till they reach the ‘Tipping Point’where Revenue < Cost of production(service).

• This cycle of price rise and crash continues.

Note that: Crash Point ≠ Tipping Point.Crash Point is reached when Revenue decreases beyond Cost of production due to Extremely Low Price.Tipping Point is reached when Revenue decreases beyond Cost of production due to Extremely High Price and Extremely Low Demand.

©Rupakshi Bhatia

The Cycle of Price Changes

Crash Point

Tipping Point

Prices FallPrices RiseTipping Point

Crash Point

Price

TimeO

©Rupakshi Bhatia

CONCLUSIONS

• The least lengths of paths that one would have to travel from one city to another vary from 1 to 3 which can be found using the Path Matrix.

• Consumers choose the flight with the greatest value, according to their Utilities of Time, Money and Comfort.

• A cycle of price changes continues in the Airline Industry with time, where the prices fall till a Crash Point, after which they start rising and rise till a Tipping Point, and then start falling again.

• At a given point in time, Indigo practices three kinds of Price Discrimination wherein the price of the tickets is determined by (i) Whether a person buys One-Way ticket or tickets for a Round Trip,(ii) The age group of the person: Infant, Child or Adult, and(iii) The time at which the ticket is being purchased.

©Rupakshi Bhatia

REFERENCES[1] Cavaliere , M; Csikasz, A; Jordan ,F. (2008 )Graph Transformations and Game Theory: A Generative Mechanism for Network FormationUniversity of Trento : 5

[2] Mankiw, G. (2012)Principles of MicroeconomicsCengage

[3] www.theincidentaleconomist.com

[4] www.eecs.umich.edu

[5] www-desir.lip6.fr

[6] www2.econ.iastate.edu

[7] www.dgca.nic.in

[8] www.goindigo.in

©Rupakshi Bhatia