This document defines and provides examples of cooperative and non-cooperative games. It discusses that in cooperative games, coalitions can form and players can negotiate binding agreements, while in non-cooperative games players act independently without negotiation. The document also defines the key elements of a cooperative game as the set of players and a characteristic function specifying the value created by different player subsets. Finally, it provides an example characteristic function for a cooperative game involving a seller and two buyers.

1. Submitted BY
Hashim.MP
NO;08
1st MA Economics

2.  A cooperative game is a game where groups
of players ("coalitions") may enforce
cooperative behaviour, hence the game is a
competition between coalitions of players,
rather than between individual players.
 An example is a coordination game, when
players choose the strategies by a consensus
decision-making process.

3.  Players negotiate binding contracts that allow
them to plan joint strategies.
 Example; Buyer and seller negotiating the
price of a good or a service or a joint venture
by two firms(eg, Microsoft and Apple)

4. A cooperative game consists of two
elements:
 a set of players, and
 (ii)A characteristic function specifying the
value created by different subsets of the
players in the game.
Formally, let N = {1, 2, . . . , n} be the (finite) set
of players,and let i, where i runs from 1 through n,
index the different members of N.

5. The characteristic function is a
function, denoted v, that associates
with every subset S of N, a number,
denoted v(S).
•The number v(S) is interpreted as the
value created when the members of S
come together and interact. In sum, a
cooperative game is a pair (N, v), where
N is a finite set and v is a function
mapping subsets of N to numbers.

6. Example
 As a simple example of a cooperative
game, consider the following set-up. There
are three players, so N = {1, 2, 3}.
Think of player 1 as a seller, and players 2
and 3 as two potential buyers. Player 1 has a
single unit to sell, at a cost of $4. Each buyer
is interested in buying at most one unit.
Player 2 has a willingness-to-pay of $9 for
player 1’s product, while player 3 has a
willingness-to-pay of $11 for player 1’s
product.

7. We define the characteristic function v
for this game as follows:
v ({1, 2}) = $9 − $4 = $5,
v ({1, 3}) = $11 − $4 = $7,
v ({2, 3}) = $0,
v ({1}) = v ({2}) = v ({3}) = $0,
v ({1, 2, 3}) = $7.

8. This definition of the function v is pretty intuitive. If
players 1 and 2 come together and transact, their total gain
is the difference between the buyer’swillingness-to-pay and
the seller’s cost, namely $5.
Likewise, if players 1 and 3 come together, their total gain
is the again the difference between willingnessto-pay and
cost, which is now $7. Players 2 and 3 cannot create any
value by coming together; each is looking for the seller, not
another buyer. Next, no player can create value on his or
her own, since no transaction can then take place.

9. Finally, note that v ({1, 2, 3}) is set equal to
$7, not $5 + $7 = $12. The reason is that
the player 1 has only one unit to sell and so,
even though there are two buyers in the set
{1, 2, 3}, player 1 can transact with only one
of them. It is a modeling choice—but the
natural one—to suppose that in this situation
player1 transacts with the buyer with the
higher willingness-to-pay, namely player3.
That is the reason for setting v ({1, 2, 3})
equal to $7 rather than $5.1

10.  In game theory, a non-cooperative game is
one in which players make decisions
independently. Thus, while players could
cooperate, any cooperation must be self-
enforcing.

11.  Negotiation and enforcement of a binding
contract are not possible
 example; Two competing firms take each
others likely behaviour into account when
independently setting pricing and advertising
strategy to gain market share

12.  For a precise formulation of a non-
cooperative game, we have have to specif
 (i) the number of players
 (ii) the possible actions available to each
player, and any constraints that may be
imposed on them,
 (iii) the objective function of each player
which he attempts to optimize (minimize or
maximize, as the case may be)

13.  (iv) any time ordering of the execution of the
actions if the players are allowed to act more
than once,
 (v) any information acquisition that takes
place and how the information available to a
player at each point in time depends on the
past actions of other players,
 (vi) whether there is a player (nature) whose
action is the outcome of a probabilistic event
with a fixed (known) distribution.

14.  we consider an N-player game, with
 N=(1,......,N) denoting the Players set

15. END