Module 9
Bargaining
What is bargaining?
• In general, bargaining situations have the following things in common:
1. The total payoff that the parties to the negotiation are capable of creating and
enjoying as a result of reaching an agreement must be greater than the sum of the
payoffs that they could achieve separately – i.e. the negotiation should create a
surplus payoff.
• Note that without a surplus payoff, there is no incentive for players to
negotiate. The existence of a surplus means players will be incentivized to
bargain.
2. Although players are bargaining to gain at the other player’s loss, this is not a zero-
sum game, as the outcome where no agreement is reached results in no payoff to
either player, and generally players will negotiate to avoid this negative outcome.
Who has the power?
• Until the advent of game theory, it was believed that a rigorous theory of bargaining
was impossible because theorists observed large discrepancies of outcomes amongst
players engaged in similar interactions.
• Theorists suggested that the ultimate division of the surplus was down to differences
in a nebulous concept of “bargaining power”.
The NE of a bargaining game
• The NE concept is not all that helpful when analysing bargaining games.
• To understand why this is the case, imagine 2 players are bargaining over R100 and
the following rules apply:
1. Both parties must simultaneously propose the share of the R100 they want for
themselves (in increments of R1)
2. If the proposals sum to R100 or less, then they each take what they proposed;
however, if their proposals sum to more than R100, then they each get nothing.
, • Note that in this game, every pair of proportions that sums to R100 will be an NE.
• This isn’t a very interesting or useful result because it doesn’t tell us anything about
how we expect the game to play out.
• Although perhaps we could predict that players may aim for mutually
predictable focal points – a 50:50 split, for example.
• Either way, having multiple equilibria raises the equilibrium selection problem, and is
just a formal way of showing that we can’t formally predict the outcome of
bargaining games.
Well, now what?
• We have just seen that the NE concept that we have devoted much of ECO2007S
towards studying doesn’t work cleanly in the context of bargaining.
• So, where do we go from here?
• There are two vastly different approaches to bargaining theory that have developed,
and we will touch on them both.
• These are:
1. Bargaining as a cooperative game
2. Bargaining as a non-cooperative game with alternating sequential offers
, PART 1:
Nash’s Cooperative Bargaining Solution
Cooperative vs non-cooperative GT
• All the games you have seen in ECO2007S so far have been non-cooperative games.
• This means that there does not exist a binding agreement between players to
ensure joint action.
• In other words, players will choose their strategies individually and will thus
choose their actions in accordance with their own self-interest.
• Ultimately, non-cooperative games assume that individuals always act to optimize
their payoffs in any subgame, regardless of the benefits that could accrue from
acting in a different manner.
• However, von Neumann and Morgernstern dedicated about half of their original
game theory book to what we call cooperative game theory.
• Cooperative game theory assumes that there is a binding agreement between
players or that players decide and implement their strategies jointly.
• We can assume that players can promise to abide by agreements and
promises once bargaining is concluded, even if doing so does not represent
the individual’s best response in the post-bargaining game.
Cooperative game theory
• We can imagine that the parties have a legal system in place that will enforce their
agreements once they sign a contract.
• This accounts for the binding agreement aspect of the cooperative game
theory framework.
• It is within this framework that Nash developed his theory of bargaining, and which
we will investigate now.
Bargaining
What is bargaining?
• In general, bargaining situations have the following things in common:
1. The total payoff that the parties to the negotiation are capable of creating and
enjoying as a result of reaching an agreement must be greater than the sum of the
payoffs that they could achieve separately – i.e. the negotiation should create a
surplus payoff.
• Note that without a surplus payoff, there is no incentive for players to
negotiate. The existence of a surplus means players will be incentivized to
bargain.
2. Although players are bargaining to gain at the other player’s loss, this is not a zero-
sum game, as the outcome where no agreement is reached results in no payoff to
either player, and generally players will negotiate to avoid this negative outcome.
Who has the power?
• Until the advent of game theory, it was believed that a rigorous theory of bargaining
was impossible because theorists observed large discrepancies of outcomes amongst
players engaged in similar interactions.
• Theorists suggested that the ultimate division of the surplus was down to differences
in a nebulous concept of “bargaining power”.
The NE of a bargaining game
• The NE concept is not all that helpful when analysing bargaining games.
• To understand why this is the case, imagine 2 players are bargaining over R100 and
the following rules apply:
1. Both parties must simultaneously propose the share of the R100 they want for
themselves (in increments of R1)
2. If the proposals sum to R100 or less, then they each take what they proposed;
however, if their proposals sum to more than R100, then they each get nothing.
, • Note that in this game, every pair of proportions that sums to R100 will be an NE.
• This isn’t a very interesting or useful result because it doesn’t tell us anything about
how we expect the game to play out.
• Although perhaps we could predict that players may aim for mutually
predictable focal points – a 50:50 split, for example.
• Either way, having multiple equilibria raises the equilibrium selection problem, and is
just a formal way of showing that we can’t formally predict the outcome of
bargaining games.
Well, now what?
• We have just seen that the NE concept that we have devoted much of ECO2007S
towards studying doesn’t work cleanly in the context of bargaining.
• So, where do we go from here?
• There are two vastly different approaches to bargaining theory that have developed,
and we will touch on them both.
• These are:
1. Bargaining as a cooperative game
2. Bargaining as a non-cooperative game with alternating sequential offers
, PART 1:
Nash’s Cooperative Bargaining Solution
Cooperative vs non-cooperative GT
• All the games you have seen in ECO2007S so far have been non-cooperative games.
• This means that there does not exist a binding agreement between players to
ensure joint action.
• In other words, players will choose their strategies individually and will thus
choose their actions in accordance with their own self-interest.
• Ultimately, non-cooperative games assume that individuals always act to optimize
their payoffs in any subgame, regardless of the benefits that could accrue from
acting in a different manner.
• However, von Neumann and Morgernstern dedicated about half of their original
game theory book to what we call cooperative game theory.
• Cooperative game theory assumes that there is a binding agreement between
players or that players decide and implement their strategies jointly.
• We can assume that players can promise to abide by agreements and
promises once bargaining is concluded, even if doing so does not represent
the individual’s best response in the post-bargaining game.
Cooperative game theory
• We can imagine that the parties have a legal system in place that will enforce their
agreements once they sign a contract.
• This accounts for the binding agreement aspect of the cooperative game
theory framework.
• It is within this framework that Nash developed his theory of bargaining, and which
we will investigate now.
