# How to Game the UTS Cards

The Ultimate Tennis Showdown (UTS) is a new tennis league created by Patrick Mouratoglou that is aiming to attract younger fans to tennis. When the league debuted this month, it was clear that it was taking a lot of inspiration from e-sports, introducing more gaming features like UTS cards. These cards give players the chance to change the value of some points, adding extra chances for strategic advantage. So far, the players haven’t always seemed to know how to use the cards to their best advantage. In this post, I show how to estimate the expected value of each UTS card and rank them from most to least valuable.

Among a number of radical changes to the way tennis is traditionally played, players get two of four possible UTS cards to play in any given quarter. Which two appears to be selected at random. The rules also state that the cards, when played, could last for variable number of points, though it so far seems that they have always had a duration of two points and that is the duration I will assume in what follows.

The four cards each have the potential to change the number of points a player can earn. Either by putting a higher award on some outcomes or by changing the win advantage of the point.

Although the rules for each card differ, their benefit can all be judged by the same yard stick. Namely, the expected value added. Here, the expected value is the expected points a player will earn over two points. The expected value added is simply how this value would change when the card is played versus if it were not.

The first we will consider is Stealing Serve. UTS is normally played in a tiebreak-like format, with players alternating serve every two points. With the Stealing Serve card a player gets to serve twice would they would normally receive, so who serves and who receives is determined by the card itself.

The expected value added, in this case, comes down to the difference in the advantage of winning a point, $P(win)$, serving versus receiving,

$$E(\mbox{Value Added}) = 2 * P(win | serving) - 2 * P(win | returning)$$

Clearly, this will vary with the particular serve/return skill of the players. The average points won on serve for top ATP players is around 65%. This means, if two well-matched tour players were facing off, this card would have an $EVA = +0.60$.

There is one other card that specifically targets the serve. This is the Serve -1 card, or no second serve card. Playing this card forces your opponent to serve without any second serve safety net.

Assessing the value of this card requires some consideration of what the opponent will do in this situation. If they are rational (which might be a big if in an exhibition format) they would play the serve like a second serve. In this case, the EVA for the card comes down to the difference in winning on return when facing a second serve versus a two-serve point,

$$E(\mbox{Value Added}) = 2 * P(win | 2nd return) - 2 * P(win | return)$$

We can look at the stats on second serve return points won to get a sense about the advantage for the average player. Among ATP players competing in the top events on the traditional tour, this is 48%. Comparing that to the average chance of 35% for winning a return point when a server has both serves suggests an EVA of $+0.26$.

The two remaining cards are interesting in that they do not specify which player is serving or receiving, which gives another layer of strategy. The first we will consider is $Win in 3 Shots$. This is another example of putting the pressure on your opponent because, with this card, they have to win in 3 shots to win the next two points.

The advantage in this case all comes down to that subset of points that your opponent wins but only in a long rally. If we assume that the number of shots is largely independent of who is serving, the EVA can be broken down as follows,

$$E(\mbox{Value Added}) = 2 (1 - P(win | return)) * P(\mbox{ends in} > \mbox{3 shots})$$

When the card is instead played when the opponent is receiving (the card player is serving), it becomes,

$$E(\mbox{Value Added}) = 2 (1 - P(win | serving)) * P(\mbox{ends in} > \mbox{3 shots})$$

Now, as has been much touted by some ‘numbers’ guys in the sport, the majority of tennis points end in 3 shots or fewer. In fact, roughly 70% of points are this short. This suggests an EVA when the opponent is serving of +0.39 and when they are receiving of +0.21. So an advantage either way but would generally be a better play on your opponent’s serve as this effectively takes away some of your opponent’s serve advantage.

The final card is perhaps the most interesting. This is the Winners x 3 card. This is the only card in the group that changes the point value of points. But, the value only changes when a player wins a point with a clean winner. If we can again treat the chance of a point ending in a winner as independent of who is serving, then a reasonable estimate of the cards added value is,

$$E(\mbox{Value Added}) = 2 * 3 * P(win | serving) * P(winner)$$

for the case when the player who plays the card is serving, and,

$$E(\mbox{Value Added}) = 2 * 3 * P(win | receiving) * P(winner)$$

in the case when the player is receiving. At Grand Slams, where we have the most stats available about winners, 30% of points end with a clean winner. This suggests an EVA for an average top player of +1.17 when serving and +0.63 when receiving.

With these summaries based on the stats for average top players, we can rank the value of the UTS cards from highest to lowest according to the situation in which they are played.

UTS Card Card Player Serves? EVA
Winners x 3 Serves +1.17