FIRST HALF

**Fernando Llorente** faces a **hard decision**. He wants to **play** next year in the **Premier League**, and there're **2 clubs interested** in signing with him: **Tottenham Hotspur F.C.** and **Aston Villa F.C.** **Fernando** likes both teams equally.

Both clubs want a **quick answer**, as they **need** to incorporate urgently **a forward** to their teams. The **conditions** of both contracts will be exactly **the same**, **except** for the **amount** of the annual **fee** of the player.

vs. |

The problem states like this: **Tottenham** will make its **offer** on **Friday**, and they want **Llorente**'s **answer** on the **same day**. **Aston Villa** will make its **offer** on **Sunday**, **without knowing** which **decision** the player has taken about **Tottenham**'s **offer**.

That way **Llorente** will have to **decide** on **Friday** which team he's going to join, without knowing if **Aston Villa**'s offer will be better or worse than **Tottenham**'s.

Apparently, whatever decision **Llorente** takes, he has a **50% chance** of **succeeding** in his decision, and a 50% of taking a wrong decision.

Let's see the **possible scenarios** that can occur:

**1.** He **accepts** **Tottenham**'s offer, and **Aston Villa**'s offer is **lower**.

**2.** He **rejects** **Tottenham**'s offer, and **Aston Villa**'s offer is **lower**.

**3.** He **accepts** **Tottenham**'s offer, and **Aston Villa**'s offer is **higher**.

**4.** He **rejects** **Tottenham**'s offer, and **Aston Villa**'s offer is **higher**.

As we can see, **Fernando** **succeeds** with his decision in **2** of the 4 cases that can occur (**1** and **4**), and he's** wrong** in the other **2** (**2** and **3**).

We dismiss the (**5**) case, which would occur if the **offers** of **Tottenham** and **Aston Villa** are **equal**, since the possibility of ocurrence is very small, and besides, whatever his choice, we can say that **Llorente** is always **right**, because he doesn't want to sign specially for any of the clubs.

If there is **no** way to know **in advance** which **offer** **Aston Villa** will make, do you think there is any way to **increase** the **probability** that **Llorente** **succeeds** with his decision?

As we have seen, the **probability** that **Llorente** **succeeds** in his choice is **50%**. Apparently, it doesn't seem to exist **any strategy** to **increase** that **success rate**. And yes, there is one. Let's see how:

In this case, the **solution** is very **rational**, and is given by what our **parents** or **friends** would advise to us: the first thing **Llorente** should do is to think how much he **wants to earn** the next seasons. And from there, he can take his **decision** in a **more accurate** way, also from a **mathematical** standpoint.

So, if Friday's **Tottenham** offer **exceeds** his claims, he should **accept** it, and if it's **lower**, he should **reject** it and sign for **Aston Villa**.

Let's see now which **possibles cases** may occur:

- If **Tottenham** **offers more than** to his **claims**, and he **accepts**:

**1.** On Sunday, **Aston Villa** **offers more money** than **Tottenham**.

**2.** **Aston Villa** **offers less** than **Tottenham**, but **more** than his **claims**.

**3.** **Aston Villa** **offers less** than **Tottenham**, and **less** than his **claims**.

- If **Tottenham** makes a **lower offer** than his **claims**, and he **rejects** it:

**4.** On Sunday **Aston Villa** **offers him less money** than **Tottenham**.

**5.** **Aston Villa** **offers more** than **Tottenham**, but **less** than his **claims**.

**6.** **Aston Villa** **offers more** than **Tottenham**, and **more** than his **claims**.

Thus, we can see that **Llorente** will **succeed** in **4** of the 6 cases (**2**, **3**, **5** and **6**), and will **be wrong** in only **2** of them (**1** and **4**).

(Here we also ignore the case in which the **offers** are **equal**, because as we saw before it was a very unlikely event and also benefited Llorente whatever his choice was).

With this **strategy**, therefore, we **increase** the **probabilities** of succeeding from **2/4** to **4/6**.

So **Llorente** simply has to think on **which amount** he wants to earn, and act according to it in order to get a **higher probability of success**.

Let's have a look at it from a **mathematical** point of view.

We have **2 offers**, one higher than the other, which we will call **'A'** and **'a'** ('A' the greatest, 'a' the smallest). And we have the economic **claims** of **Fernando Llorente**, that we'll call **'F'**.

The only **possible cases** that may occur, knowing that **A>a**, are:

**1)** A > a > F

**2)** A > F > a

**3)** F > A > a

**4)** A = a

As mentioned, **Fernando Llorente** will accept **Tottenham**'s offer if it **exceeds** the amount he wants to earn.

In the first case (**1**), as **both offers** are **above** **Llorente**'s expectations, he will choose the first offer, and will join **Tottenham** team, so he will be **wrong** **50%** of the time (sometimes the highest offer **'A'** will be **Tottenham**'s, and sometimes will be **Aston Villa**'s).

In the second case (**2**), when **an offer** **exceeds** **Llorente**'s expectations, and the **other offer** is **lower**, **Fernando** will **always succeed**, because if **Tottenham**'s offer is **'A'** he will accept it, and is it's **'a'**, he will reject it and accept offer **'A'** from **Aston Villa**.

In the third case (**3**), **both offers** are **below** **Llorente**'s expectations, so he will not pick the first one, and he will join **Aston Villa**, with a success of **50%** of the time (as in first case, sometimes the highest offer **'A'** will be **Tottenham**'s, and sometimes will be **Aston Villa**'s).

In the latter case (**4**), **both offers** are **the same**, and as **Llorente** doesn't like one team more than the other, we can say that he **always succeeds**, whatever his choice).

If we call:

**P(C1)** = probability that both offers exceed Llorente's expectations, that is, **A>a>F** (case 1)

**P(C2)** = probability that Llorente's expectations are among the 2 offers, that is, **A>F>a** (case 2)

**P(C3)** = probability that both offers are below Llorente's expectations, that is, **F>A>a** (case 3)

**P(C4)** = probability that both offers are equal, whter they are above or below the claims of Llorente, this is, **A=a** (case 4)

Necessarily one of the 4 cases must happen, so we have:

If now we call **P(x)** to the **overall success probability**, we have that it's eual to the success probability in each of the 4 possible cases:

As we have seen before, in cases **C1** and **C3** the success probability is **50%**, and in cases **C2** and **C4** Llorente succeeds always, **100%** of the time:

Since there is always a **chance** that Llorente's **claims** are **among both offers**, in other words, as **P(C2)** is always **greater than zero**, and as the probability that both offers P(C4) is also greater than zero, then the overall success probability P(x) is always greater than 1/2:

We conclude, therefore, that **with this strategy** **Fernando Llorente** will improve his chances of obtaining a **more favorable contract**.

This **problem** consists in **choosing the highest number** among two hidden numbers, that we discover one by one. This is a **paradox** in which, unlike other ones, we get the **best result** by applying our **common sense** and solving it by the way we often face this type of situations (or at least, by the way we should face them).

As we **increase the number of available options**, we must follow the **optimal stopping theory**, which consist in determine the **optimal moment** to take a **decision**, in order to **maximize** the **profit**, stablishing an optimal **stopping** **rule**. We find such kind of problems in different areas like statistics, finances and stock market, etc., as in our everyday life. It's about **decisions** we have to take on either **accepting** an offer at a precise **moment**, with no posibility of resuming a last offer, and without knowing which offer we'll have in the future.

The **optimal stopping theory**, performed among others by **Franz Thomas Bruss**, from the Free University of Brussels (Université Libre de Bruxelles), who found out **the 1/ e-law of best option**, that we can use for solving lots of

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