Abstract your initialization into a method, and call the method from mounted and wherever else you want.
new Vue({
methods:{
init(){
//call API
//Setup game
}
},
mounted(){
this.init()
}
})
Then possibly have a button in your template to start over.
<button v-if="playerWon" @click="init">Play Again</button>
In this button, playerWon represents a boolean value in your data that you would set when the player wins the game so the button appears. You would set it back to false in init.