Rock, Paper, Scissors and Trump

Question

Alicia and Robert are two high-stakes gamblers who are playing a match of rock-paper-scissors. Each winning throw is worth 10 €, paid by the other player. These players have deep pockets, so matches can last a long time and proper strategy is important.

To make things interesting, Alicia and Robert have added a special rule to the game. Each player selects a trump, which is a throw that nets them an extra 5 € every time they play it, regardless of whether they win, lose or draw with that throw. Alicia has selected rock as her trump, and Robert's trump is paper.

Examples:

Alicia throws scissors, Robert throws rock. Robert wins 10 €.
Alicia throws paper, Robert throws paper. Robert wins 5 €.
Alicia throws rock, Robert throws scissors. Alicia wins 15 €.

Can either player win this game in the long run? What strategy should they use?

Answer 1

Looks like

Robert can win.

One strategy would be to

on each throw, randomly choose between two rocks, three papers, and one scissors. (Secretly roll a 6-sided die to decide.)

The opponent will naturally soon realise what's happening. But it won't help. Here are the possible results from Alicia's point of view:

    \ Robert   R    R   P    P    P    S | Total |
   Alicia +------------------------------+-------+ 
     Rock |   +5   +5  -10 -10  -10  +15 |   -5  |
    Paper |  +10  +10  -5   -5   -5  -10 |   -5  |
 Scissors |  -10  -10  +5   +5   +5    0 |   -5  |
          +------------------------------+-------+
 

So whatever the opponent chooses, the expected value will favour the strategy given above. By the law of large numbers, the average amount gained from a single throw

should gradually approach five sixths of an Euro (pretty close to one US Dollar, actually) as the game goes on.

(Thanks, @Adayah, for pointing out that I hadn't actually included this number in my original answer)

The reason this game isn't symmetrical has to do with one trump beating the other.

The way I went about solving this puzzle was to first notice that the trump asymmetry

must give Robert an advantage, because both players would like to play their trumps often, but Robert's trump getting played a lot makes it unprofitable for Alicia to play her trump. This means that Robert is in an arbitrage position: all he has to do is to bet "mostly paper", and hedge his bets by betting against himself a bit (choosing anything else than paper is bad for Robert, assuming Alicia chooses randomly), so that his advantage "spreads" to all possible plays by Alicia.

This bet hedging is extremely important, because in arbitrage betting, you can occasionally get away without making sure you win at every possible outcome. In a game of repeated RPS, however, there is an intelligent adversary, and in game theoretical calculations, this guarantees that you will always hit the worst possible outcome with any strategy. Therefore, the only thing you need to optimise is the outcome in the worst case scenario. (For more details and info on the subject, see the excellent comments by Gareth and Jaap below.)

The fact that the first suitable betting strategy I stumbled upon happened to spread the advantage evenly (making the worst case equal to the best case), and with minimum loss of advantage, speaks volumes for the excellent design of the puzzle. Thanks, @jafe!

Answer 2

A/R	R	S	P	±
R	+5	+15	-10	+10
S	-10	0	+5	-5
P	+10	-10	-5	-5
∓	-5	-5	+10	0

This table shows Alicia's return for each game. Reading as negated gives Robert's.

The first choice for A would then be Rock, but R's trump beats this, and once this is known, Rock is a useless choice for A, and to stop R playing trumps, she should choose Scissors.

But this leads to a cat-and-mouse game, with no discernible strategy, so no determinable winner.

Answer 3

I think Robert would win if he keeps throwing his trump.

The cases are

1. Alicia throws Rock (5e) < Rob's paper (15e)

2. Alicia's Scissors(15e) > Rob's paper (20e)

3. Alicia's Paper (15e) = Rob's Paper (25e)

We see that if Robert sticks to paper only then no matter what Alicia chooses, he will be in the lead.

Now if Alicia follows the same and throws her trump i.e rock then Robert will still be in the lead because his trump is better than hers.

Now let's consider the case of Alicia throwing only scissors. If she catches on to Rob's strategy then she will throw only scissors which is the second case.
But Rob will always have a 5e advantage and all he needs to do is switch to rock when he feels there is a chance that Alicia will chose scissors twice in a row.

Answer 4

Just like the original RPS,

There is no answer.
So, Alicia might think that Robert would use his trump, and so would play scissors, but Robert knows this, and would play rock instead, and you could go endlessly down the rabbit hole, though there is no way to have a 100% chance to beat a random order. But, Robert will play his trump more, but Alicia won’t as much, because she knows that is they both use their trump, Rob will net 10, but Alicia can net 15 if Rob plays scissors to her trump rock.
Therefore Rob will generally play scissors less, and also rock less, because if they both play rock then she nets 5, and his trump more, which would make her more likely to use scissors, and eventually this will contradict itself, because no matter how deep you go into the rabbit hole, you can’t get a strategy that will give you an advantage.