"An encoder is a device, circuit, transducer, software program, algorithm or person that converts information from one format or code to another, for the purposes of standardization, speed or compression."
You are converting information from one format to another. But it's the score of the game. In this case your entire game is the entire system. In my opinion an encoder is a part that can be put in between two systems. So I'd say that... it's in a really gray area but it leans towards not being an encoder.
I'd call these things for encoder:
- Rotary encoder, it's bridging information between something that rotates and some microcontroller. This is important.
- Forward error correction codes, it's about converting some bits of data to some other bits that you are then sending through a medium, like wireless through the air. And then the receiver change the bits that was sent back to the data that was encoded on the sender side. Without this encoding it would take you minutes to load this page. This is important.
- UTF-8, it's one of many text encodings. This one allows me to write the Swedish "åäö" and some Japanese "きみ わ". This is important.
It's like a human translator allowing someone who's deaf to communicate with someone who's not deaf but doesn't know sign language.
In your case there's... gahh.. this is so gray. There are three systems, you, your opponent, and the game. But no, my list above that shows what I'd call an encoder are vital parts of the system. Or things that matter really much, without those encoders the systems have to use other much more difficult alternatives. Your game can be played without the hardware, which might be even simpler. Your hardware version of the game is not important.
An encoder shouldn't be the main focus of a system, it should be something that another system can use for some positive effect. Your game is the main focus.
So no, I stand my point as classifying your game as not being an encoder.
I'd call it what I've done throughout my answer most of the time. A game. But if you want something more nerdier then I'd call it a "decider" circuitry, since neither you nor your opponent will decide who's the winner, but the circuitry will, and both of you will take it at face value. Or I'd call it "game circuit" since that is a pretty correct description of what it actually is.