TECHNICAL

The games people play

Michael Schumacher, Brazilian GP 2000

Michael Schumacher, Brazilian GP 2000 

 © The Cahier Archive

Before pit stops to refuel and change tires were reintroduced into Formula 1, there was only one race strategy, as Colin Chapman told me at my first race as a race engineer: "Go like hell at the start, and overtake as many people as possible." Nowadays the post-race talk is as much about strategies as actual car performance or on-track overtaking maneuvers.

Part 1.

The Greeks coined the term strategy, defining it as "generalship" or the art of war, and it has been widely used and misused ever since to describe such highly competitive and often combative activities as economics, business and motor racing. What the Greeks probably did not anticipate however was that strategy would be turned from an art into a science, and that computers would enable that science to be practically applied to many situations.

In 1928, John von Neumann, one of the great mathematicians and versatile minds of the last century, published his first paper on Game Theory. However, it was not until 1944, when his Theory of Games and Economic Behaviour was published, that the world started to take notice. A reviewer commented on the work: "Posterity may regard this book as one of the major scientific achievements of the first half of the twentieth century." Although the theories had been developed as a basis for economics, it was the military that first really grasped the concept and saw how it could be applied to the mind-boggling task of waging war on the scale that had emerged during WWII.

War, economics and even a Formula 1 race, are extremely complex situations, and thus difficult to analyze. What von Neumann realized was that they have much in common with many games (there is no evidence that he included Formula 1 in his deliberations), and that studying how people play games and developing a theory of how players set about attempting to win them, would provide a basis for the analysis of more complex interactions between humans in conditions of conflict. The game most often used as an example of Game Theory is Poker, and if the reader is sceptical that war could ever just be an extension of a game of Poker, let us look at the essential elements of the game and at the same time, watch out for parallels between Poker and war (and while we are about it, a Formula 1 race).

Around a poker table there are seated a number of players. The first step is to identify the number of sets of opposing interests, which may not necessarily be the same as the number of players. Coalitions may exist, some open, some secret, some false. Each player (or coalition) wants to win, and because their winnings are the losses of others, their interests are opposed - that is the basis of conflict. Some of the elements of the action are by personal choice and completely within the individual control of the players. Because this is true for each player, some of the elements are out of a player's control and, what is worse is that these elements are controlled by minds that have opposing interests. Then there are elements that, according to the rules, are not within any players' control, e.g. the order of the cards in the deck. These latter elements can be considered as being controlled by Luck - whose character is stable if somewhat puckish, but is not consciously malicious.

A player's state of information is also an important factor, and is subject to unreliability. Bluff and counter bluff - the distribution of false information regarding strengths and intentions - tend to confuse the intelligence gathering process. The rules are not always adhered to and, just occasionally, someone gets killed.

The essence of Game Theory analysis is to identify the opposing interests in the game, all the possible strategies usable by each one, and the payoffs (usually a transfer of assets) that are a consequence of each and every combination of strategies. To illustrate, I will use an even simpler game than Poker, one that has been played by children of all ages, for many centuries: Scissors-Paper-Stone. There are two players, each of whom makes their closed hand, on a count of three, into a symbol of scissors, stone or paper. Scissors cut Paper; Paper wraps Stone; Stone breaks Scissors, and thus the winner is determined. The analysis is portrayed as a matrix of the three possible strategies of each of the two players. The pay-off is scored as 0 for a draw, +1 for a Player A win, -1 for a Player B win, i.e. it tracks the flow of assets from Player B to Player A.

Player A

Scissors

Paper

Stone

Scissors

0

-1

1

Player B

Paper

1

0

-1

Stone

-1

1

0

Various things are apparent from this simple analysis. There is no dominant strategy, guaranteed to maximize winnings or minimize losses. Each strategy has the same odds of winning, and so should be played randomly. A player should not play to a system, as this may become detectable to an intelligent opponent, especially if it is based on the opponent's previous play. In this two player, three-strategy game there are 9 possibilities.

It can also be seen that as the number of players increases and the number of possible strategies goes up, the number of combinations increases dramatically. Calculating the payoffs, especially if they are not a simple 0,+1 or -1, requires serious number-crunching, and is the province of parallel-processing supercomputers - or an agile mind. In Chess, where the number of players is just two, but the number of strategies is very large, supercomputers have only just developed to the stage where they can out-compute the best chess-minds. Game Theory as applied by the military is one of the applications that has pushed the number-crunching capabilities of computers to the performance they have today, and will continue to do so.

Development of Game Theory beyond the simple example above is way beyond the scope of this article and involves a great deal of mathematics. For those who wish to pursue the subject further, I would recommend The Compleat Strategist by J.D.Williams as the essential primer on the subject.

An example of how Game Theory can be applied to engineering design problems, and the enormity of the computational task, can be found in an SAE paper presented at the 2000 Motorsports Engineering Conference:

2000-01-3564 Racecar Optimization and Tradeoff Analysis in a Parallel Computing Environment. Kurt Hacker and Kemper Lewis, State University of New York at Buffalo; Edward M. Kasprzak, Milliken Research Associates, Inc.

The authors describe techniques for computer optimization of racing cars. Taking just three parameters: aerodynamic downforce distribution, weight distribution and roll stiffness distribution, they model the performance of a car on two skid pads of different radii, and analyze the trade-offs between these parameters to determine the optimum solution using "Pareto analysis" - a Game Theory technique. The task is very computer intensive, requiring 18,496 simulations, and the authors describe how it can best be carried out with a parallel computing architecture. 32 Sun Ultra 5 workstations were employed, configured as a supercomputer.

Enough of the theory. In Part 2, I shall use the example of David Coulthard's Austrian GP win to examine whether it was McLaren's "brilliant strategy" that enabled Coulthard to beat the Ferraris, or whether Luck intervened, and how Formula 1 teams may use Game Theory to assist them in devising race-winning strategies. Perhaps they already do.

Part 2.

In Part 1, I described an approach to strategising based on Game Theory. While this computer-intensive approach is widely used by economists and the military, Formula 1 teams are just beginning to use computational assistance in devising race strategies. David Coulthard's Austrian GP win was hailed as being, partially at least, due to McLaren's "brilliant strategy". Was it?

While not as complex as the economy or war, a Formula 1 race is more complex than a game of Poker, due to the number of players involved and the number of possible strategies available. If we just deal with strategies during a race, there are 22 players and a large number of options for planning and executing refuelling stops. There are fewer than 22 sets of opposing interests, as some drivers within a team cooperate due to team orders, contractual obligations or because they agree to, and it has even been known for Teams to cooperate. It is the refuelling/tire change pit stop that is the crux of modern race strategy, particularly as it offers one of the few opportunities for overtaking. The decision that has to be made prior to the race is how much fuel to put in the tank. This does not fix the lap during which the driver will stop, but creates a window of stopping opportunities, though even this may change based on actual fuel consumption.

What is the optimum time to stop? Intuitively one would say, the middle of the race.

LAP

FUEL

LAP

Fuel

LAP

Fuel

LAP

Fuel

LAP

Fuel

1

1

1

2

1

3

1

4

1

5

Pit Stop

2

1

2

2

2

3

2

4

2

5

Pit Stop

3

1

3

2

3

3

3

4

3

4

Pit Stop

4

1

4

2

4

3

4

3

4

3

Pit Stop

5

1

5

2

5

2

5

2

5

2

Pit Stop

6

1

6

1

6

1

6

1

6

1

Total

16

Total

13

Total

12

Total

13

Total

16

The above tables show the pit stop options in a simple 6-lap race (assuming one had to stop) stopping after the 1st, 2nd, 3rd, 4th and 5th laps, and calculating the fuel burden (in lap's worth, rather than liters) to be carried for each of the options. Stopping after 3 laps is the optimum. It is probably also optimum for tire performance and the effect on handling of the CG height, but lets keep it simple - we don't need a computer yet, the back of an envelope will do so far. The teams will have calibrated the car on different fuel levels and tire options during testing and in Practice on Friday and Saturday morning, and have all the curves for lap time versus fuel, lap time versus laps on tires, fuel consumption versus fuel load and lap time, etc. in their computers, and be able to crunch the equivalent tables for a 70+lap race.

The tank capacity of individual cars is a well-preserved secret, but is around 160 liters. A lap of the A-1 Ring will consume around 3.5 liters, weighing around 2.6 kg, and each lap's worth of fuel will slow the car by around 0.05 seconds per lap. The 71-lap race will therefore require 250 liters (185 kg) of fuel, and a 160-liter tank will offer a window for a pit stop from lap 25 to lap 46 (Austria is a one stop race), at normal fuel consumption rates. These figures are round number guesstimates, as I do not of course know the actual figures. If the fastest way to complete the race is to stop on lap 35 or 36, why did Schumacher stop on lap 46, Barrichello on lap 47, and Coulthard on lap 50? Who had the best strategy and why was it not the calculated optimum one?

The obvious answer is Coulthard, as he won the race. None of the top three finishers started the race with the apparently optimum fuel load, and Coulthard was able to go 4 more laps than Schumacher before stopping. He either started with 4 more laps of fuel in the tank, or the Mercedes engine was more frugal than the Ferrari, or a combination of the two. If we take the case where Coulthard started with just 2-more laps of fuel than Schumacher, we can analyze various scenarios and we will see why starting the race with more fuel than optimum can often make sense. Let us also assume, initially, that it was a straight race between Schumacher and Coulthard, with no distractions.

For the first 46 laps, Schumacher would have had 7 liters (5.2 kg) less fuel in his tank, worth 0.1 seconds per lap, and so he would have pulled out a lead of 4.6 seconds. Once he had refuelled, he would have had 23 laps' worth more than Coulthard (80.5 liters, 60 kg) for the 2 laps before Coulthard came in. Coulthard would have been 1.15 seconds per lap faster, and gained 2.3 seconds, but would have still ended up 2.3 seconds behind Schumacher. So, in a straight race, the nearer the optimum starting fuel load, the better. For Schumacher, starting from Pole, a lighter fuel load than Coulthard was the best strategy; but why not the optimum? If we look at what actually happened, we can see why.

McLaren had a disastrous Qualifying, suffering some sort of collective brain fade, and their two men ended up 7th and 8th on the grid, from whence they could expect themselves to be well boxed in at the start. The Warm-up on Sunday morning confirmed that there was nothing in fact wrong with the cars, and therefore their initial race speed would be dictated by those around them, rather than by their car's actual potential. Putting extra fuel in the cars would probably not slow them up. At the start Luck, in the new guise of Launch Control (what on earth has McLaren done to deserve Luck deserting them so often recently?) and displaying it's puckish nature, this time played into Coulthard's hands. Coulthard found himself 5th, just two places behind Schumacher, who was unable to extend his Ferrari to its full potential due to the two Williams-BMWs ahead. Schumacher's battle with Montoya was so fierce, as the latter struggled with fading Michelins to keep Schumacher back, that Coulthard was able to lean off his Mercedes engine to save fuel and cruise around watching the pair ahead holding each other up and waiting for them to have their inevitable coming together. The weight penalty of however much more fuel he had started with was not resulting in him being left behind by the leaders - which after the dust settled turned out to be just Barrichello. In the event, he was able to stay out for three more laps than Barrichello, and gained the necessary seconds during those three laps to effectively overtake him during the pit stops. Ferrari, in anticipation of such possible events, had tried to give themselves the widest possible options by putting as much fuel as possible into their cars, but whether the McLaren has a bigger tank, or the Mercedes is more frugal, or whether Montoya's pace allowed Coulthard to lean off enough to save sufficient fuel, McLaren's strategy was the right one under the circumstances.

How extensively McLaren and Ferrari, and any other Formula 1 Teams use computers and strategic analysis techniques such as von Neumann's Game Theory, to determine race strategies, is not clear. During races, such as in Austria, playing out those strategies as the circumstances of the race develop, and the other opposing interests and Luck play their hands, requires both information, and either massive computing power or agile and intuitive minds. Computing some of the simpler scenarios, such as the effect of all the other runners' choice of possible pit stop strategies on one's own, are feasible. Teams have invested in enormous processing power for CFD, so computing power should not be a problem. A key feature of determining the optimum time to stop, is determining where the driver will come out after the stop, among other slower cars. The information source for this is the Heuer timing and Seimens information systems, which is fed to the Teams as an analogue TV signal. This is not easily input into computers (the banks of TV's on the Pit Wall are TV displays only, not computer screens), and so cannot be automatically used in strategic analysis. It is inevitable that this data will eventually become available as a digital data stream, with all the Teams on a network linked to the Heuer-Seimens system. This data is a major part of the TV "product" of Formula 1, and so is not yet widely and freely available in a form that can be used by those that have not paid for it. Once the Teams get access to digital data about the position and speed of all the cars, only processing power will limit what can be analyzed in virtually real time.

Strategic planning in Formula 1 is not of course limited to races. Every aspect of running a Team requires long, medium and short term strategic planning, and the tools that have been developed for the military and economists are and will be applied more and more. Ross Brawn has described Ferrari as being in a state similar to being at war. In Austria, General Brawn lost that particular battle, more due to circumstances than a bad strategy, and Field Marshall Todt, whose responsibility it is to win the war, ordered Captain Barrichello to let Major Schumacher through. We will have to wait till later in the season to see if that was the right strategy.

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