Who wins the contract talks? Bargaining, walk-away offers and deadline day

A player and a club both gain from a new deal; the only question is how they split the gain. Game theory's answer is that it depends on what each could get by walking away, and on who's in more of a hurry.

Beginner Part 8 of Game Theory Through Football

New to the notation? The symbols explained

Contents

The football question
The concept
The talks in numbers
Change the walk-aways, change the wage
Why a year left changes everything
Patience is power
Why it matters
Limitations
Try it yourself
Reproduce the analysis
Further reading

The football question

A club wants to keep its best player, and the player is happy to stay. Both sides gain from a new contract. So what should his wage be, and who decides? Agents talk tough and clubs leak stories, but underneath there's a simple question: how do two sides split something they can only get by agreeing?

The concept

Every negotiation has a deal zone. At one end is the least the player would take: his best offer from anywhere else. At the other is the most the club would pay before it would rather walk away and use its next-best option. Any wage in between leaves both better off than no deal.

In 1950 John Nash, the same Nash as the penalty equilibrium in part 1, asked which wage in the zone two sensible sides should settle on. His answer, the Nash bargaining solution, is the wage that makes the two sides' gains, multiplied together, as big as possible. When money is all that matters, that's a simple rule: split the difference. Each side gets what it would have had by walking away, plus half of what the deal adds.

The talks in numbers

The numbers are made up, to show the idea, in £m a year:

The player is worth 10 a year to the club, in what he adds on the pitch and off it.
His best other offer is 3.
The club's next-best option, signing someone cheaper, leaves it 4 a year better off.

The player won't take less than 3. The club won't pay more than 6, because above that it's better off with the replacement (10 − 6 = 4). So the deal zone runs from 3 to 6, and the deal adds 3 on top of both walking away. Splitting the difference puts the wage at 4.50.

$$w = d_p + \frac{(V - d_c) - d_p}{2}$$

In plain football

V is what the player is worth to the club each year, here 10.
d_p is the player's walk-away: his best other offer, 3.
d_c is the club's walk-away: what its next-best option leaves it, 4.
V − d_c is the most the club would pay, 6. The gap between that and 3 is what the deal adds, and the wage w gives the player half of it: 3 + 1.50 = 4.50.

Notice what doesn't decide the wage: who shouts loudest, or what the player "deserves". Only the two walk-away positions and what the player is worth.

Change the walk-aways, change the wage

Each bar is a deal zone, from the least the player would take to the most the club would pay; the dot is the Nash wage, halfway across. Made-up numbers.

A rival club offers 5. The player's walk-away rises, the zone shrinks to 5 to 6, and the wage jumps to 5.50. That's why agents go looking for other offers: an outside offer moves the floor, and the wage follows it up.
The club finds a cheaper replacement, worth 5 to it instead of 4. Now the club won't pay more than 5, and the wage falls to 4.00.
A rival offers 7. That's more than the club would ever pay, so there's no deal zone at all. The player should go, and both sides are better off for it.

Why a year left changes everything

As a contract runs down, both walk-aways move. With years left, the club can sell the player for a fee, so walking away from talks isn't so bad for it. Since the Bosman ruling of 1995, a player in Europe at the end of his contract can leave for nothing, and a free agent can often get a signing-on fee elsewhere. In made-up numbers:

Years left	Walk-aways (player, club)	Wage
Two	3, 6	3.50
One	3, 5	4.00
None	4, 4	5.00

Same player, same worth to the club, and the wage rises from 3.50 to 5.00 just because the clock ran down. That's why clubs try to agree new deals early, and why players with a year left suddenly get offered more, or get sold.

Patience is power

Nash's answer assumes both sides are equally patient. In 1982 Ariel Rubinstein looked at talks as offer and counter-offer, where every round of delay costs both sides something. His answer: the more patient side gets more.

Say the club makes the first offer, and each round of delay leaves both sides with a little less (here, 95% of what was on the table for the club). If the player is just as patient, he gets 49% of the 3 the deal adds, a wage of 4.46, very close to splitting the difference; if both are more patient still, it's 50%, 4.49. But if the player is desperate to get it done before the transfer window shuts, losing a fifth of what's on the table with each delay (80%), his share falls to 17% and his wage to 3.50.

That's deadline day in one sentence: the side that can't wait pays for it.

Why it matters

Know your walk-away, and theirs. The wage is decided by the alternatives, not the arguments. Improving your own walk-away, or learning theirs, matters more than anything said in the room.
Some deals shouldn't happen. When the player's best offer is more than the club would ever pay, letting him go is the right answer for both.
Timing changes power. A contract running down shifts the deal zone towards the player, and a deadline shifts it towards whoever can wait.
It's the same game beyond football: a pay rise, buying a house, any deal where both sides gain but have to agree how.

Limitations

The numbers are made up. Clubs don't publish what a player is worth to them, and walk-away offers are often secret.
Each side may not know the other's walk-away. Real talks involve bluffing about it, the subject of part 10.
Money isn't everything. Players care about playing time, family and ambition; clubs about dressing-room harmony. Any of these changes the walk-aways.
Split the difference assumes equal footing. With more bargaining power on one side, the split tilts, as the patience example shows.

Try it yourself

Change the walk-aways in the snippet: give the player a rival offer of 6 and the club a replacement worth 3, and see where the wage lands. Or think of the last contract saga you followed: what was each side's walk-away, and did the deal land where the theory says?

Reproduce the analysis

This needs nothing but Python. All the numbers are made up.

Show the Python38 lines, ready to copy and run.

# From Football Data Science by Bryan McGuire. Free to use with credit.
# https://www.footballdatascience.co.uk/learn/bargaining-contract-talks
# Contract talks as a bargaining game. All the numbers are made up, in £m a year.
VALUE = 10   # what the player is worth to the club each year he plays for it


def nash_wage(player_walk, club_walk, value=VALUE, step=0.01):
    """Nash's bargaining solution: the wage that makes the product of both sides' gains, over walking away, largest.
    A wage only works if both gain: at least the player's best other offer, at most what keeps the club ahead."""
    low, high = player_walk, value - club_walk
    if low > high:
        return None   # no deal leaves both better off
    wages = [low + i * step for i in range(round((high - low) / step) + 1)]
    return max(wages, key=lambda w: (w - player_walk) * (value - w - club_walk))


print("player's best other offer 3, club's next-best option worth 4")
print(f"  deal zone {3:.2f} to {VALUE - 4:.2f}; Nash wage {nash_wage(3, 4):.2f}")
for why, p, c in (("a rival club offers 5", 5, 4), ("the club finds a cheaper replacement, worth 5 to it", 3, 5),
                  ("both: rival offer 5, replacement worth 5", 5, 5), ("rival offer 7, replacement worth 4", 7, 4)):
    w = nash_wage(p, c)
    print(f"  {why}: " + (f"deal zone {p:.2f} to {VALUE - c:.2f}, wage {w:.2f}" if w is not None else "no deal: the zone is empty"))

# As the contract runs down, the club's fallback (selling him for a fee) shrinks and the player's grows
# (a free agent gets a signing-on fee). Made-up fallbacks by years left on the old deal:
print("\nyears left: player's fallback, club's fallback -> wage")
for years, p, c in ((2, 3.0, 6.0), (1, 3.0, 5.0), (0, 4.0, 4.0)):
    print(f"  {years}: {p:.1f}, {c:.1f} -> {nash_wage(p, c):.2f}")

# Patience: Rubinstein's alternating offers. Each round of delay shrinks what's left to share by a factor (closer
# to 1 = more patient). The side making the first offer keeps (1 - other's factor) / (1 - product) of the surplus.
def first_mover_share(d_first, d_second):
    return (1 - d_second) / (1 - d_first * d_second)

surplus = (VALUE - 4) - 3   # what a deal adds on top of both walking away: 3
print("\nclub offers first; player's share of the 3 the deal adds, and his wage")
for why, d_club, d_player in (("equally patient", 0.95, 0.95), ("equally, very patient", 0.99, 0.99),
                              ("player wants it done before the window shuts", 0.95, 0.80)):
    share = 1 - first_mover_share(d_club, d_player)
    print(f"  {why} (club {d_club}, player {d_player}): {share:.0%}, wage {3 + share * surplus:.2f}")