Working on a Rock Paper Scissors game to learn Apex/visualforce with (and to be able to play with other devs on the org). Somewhere in the main body, something is not working right. It is running, that much I know, but somewhere down the line it is not updating the page nor the database as to the result. Here is the VF page:

<apex:page showheader="false">
<c:rps ></c:rps>
</apex:page>

Here is the Component:

<apex:component controller="rps" allowDML="true">
Lets play Rock Paper Scissors!
    <apex:form >
        <apex:selectList id="rps" value="{!input}" size="1" title="Choose Your Weapon">
            <apex:selectOptions value="{!inputSelect}"/>
        </apex:selectList>
        <apex:selectList id="rps2" value="{!partners}" size="1" title="Choose Your Weapon">
            <apex:selectOptions value="{!partner}"/>
        </apex:selectList>
        <apex:commandButton action="{!makeNew}" value="Create Game"/>
        <apex:commandButton action="{!makePlayable}" value="Allow Multiplayer"/>
    </apex:form>
    <apex:form >
        <apex:pageBlock >
            <apex:pageBlockButtons >
                <apex:commandButton value="Play" action="{!processSelected}" rerender="table"/>
            </apex:pageBlockButtons>
            <apex:pageBlockTable value="{!ListAvailable}" var="a" id="table">
                <apex:column >
                    <apex:inputCheckbox value="{!a.selected}"/>
                </apex:column>
                <apex:column value="{!a.game.Name}" />
            </apex:pageBlockTable>
        </apex:pageBlock>

    </apex:form>
    <apex:outputPanel id="dynamic">
        {!output}<br/>
    </apex:outputPanel>
</apex:component>

Here is the controller class:

public class rps{
    
    public PageReference makeNew(){
        Asynch_RPS_Game__c a = new Asynch_RPS_Game__c();
        a.move__c = input;
        a.user2__c = partners;
        INSERT a;
        
        return null;
    }
    
    public List<SelectOption> getPartner(){
        List<selectOption> op = new List<selectOption>();
        for(RPS_Save__c rps : [SELECT Name FROM RPS_Save__c WHERE MakePlayable__c = true]){
            op.add(new selectOption(rps.Name,rps.Name));
        }
        System.debug('op: ' + op);
        return op;
    }
    
    //Gives options to click
    public List<SelectOption> getinputSelect(){
        List<selectOption> options = new List<selectOption>();
        options.add(new selectOption('rock','Rock'));
        options.add(new selectOption('paper','Paper'));
        options.add(new selectOption('scissors','Scissors'));
        return options;
    }
    
    public String partners {get;set;}
    public String input {get;set;}
    public String output {get;set;}
    public integer wins {get;set;}
    public integer losses {get;set;}
    public integer ties {get;set;}
    private RPS_Save__c userScore;
    public List<aAsynch_RPS_Game> gamesList {get;set;}
    
    //allows the multiplayer
    public PageReference makePlayable(){
        RPS_Save__c[] userScores = [SELECT id FROM RPS_Save__c WHERE Name=:userInfo.getName()];
        if(userScores.size() > 0){
            userScore = userScores[0];
        } else {
            userScore = new RPS_Save__c(Name = userInfo.getName(), Wins__c = 0, Losses__c = 0, Ties__c = 0);
            INSERT userScore;
        }
        userScore.makePlayable__c = true;
        UPDATE userScore;
        return null;
    }
    
    //Lists games that still need to be finished
    public List<aAsynch_RPS_Game> getListAvailable(){
        if(gamesList == null) {
            gamesList = new List<aAsynch_RPS_Game>();
            for(Asynch_RPS_Game__c a : [SELECT Name, Owner.Name, Move__c, Finished__c FROM Asynch_RPS_Game__c WHERE User2__c =: userInfo.getName()]) {
                if(a.Finished__c == null){
                    gamesList.add(new aAsynch_RPS_Game(a));
                }
          }
        }
        return gamesList;
    }
    
    
    
    //does stuff when you select things
    public PageReference processSelected(){
        system.debug('is this even running?');
        List<Asynch_RPS_Game__c> available = new List<Asynch_RPS_Game__c>();
        for(aAsynch_RPS_Game ggame : getListAvailable()){
            if(ggame.selected == true){
                available.add(ggame.game);
                system.debug('sup from avail.add');
            }
        }
        
        integer l;
        integer w;
        integer t;
        
        RPS_Save__c[] userScores = [SELECT id, Wins__c, Losses__c, Ties__c, Name FROM RPS_Save__c WHERE Name=:userInfo.getName()];
        if(userScores.size() > 0){
            userScore = userScores[0];
          w = integer.valueOf(userScore.Wins__c);
          l = integer.valueOf(userScore.Losses__c);
          t = integer.valueOf(userScore.Ties__c);
        } else {
            userScore = new RPS_Save__c(Name = userInfo.getName(), Wins__c = 0, Losses__c = 0, Ties__c = 0);
            INSERT userScore;
        }
        
        for(Asynch_RPS_Game__c a : available){
          integer comp;
            if(a.move__c == 'rock'){
                comp = 0;
            }else if(a.move__c == 'paper'){
                comp = 1;
            }else{
                comp = 2;}
        
            //check win/loss/tie
          if(comp == 0){
              if(input == 'rock'){
                  output = 'Tie';
                  if(t > 0){
                      t++;
                  }else{
                      t = 1;
                  }
                  return null;
              }else if(input == 'paper'){
                  output = 'Win';
                  if(w > 0){
                      w++;
                  }else{
                        w = 1;
                  }
                  return null;
              }else if(input == 'scissors'){
                  output = 'Lose';
                  if(l > 0){
                      l++;
                  }else{
                      l = 1;
                  }
                  return null;
              }else{
                  output = ' ';
                  return null;
              }
          }else if(comp == 1){
              if(input == 'rock'){
                  output = 'Lose';
                  if(l > 0){
                      l++;
                  }else{
                      l = 1;
                  }
                  return null;
              }else if(input == 'paper'){
                  output = 'Tie';
                  if(t > 0){
                      t++;
                  }else{
                      t = 1;
                  }
                  return null;
              }else if(input == 'scissors'){
                  output = 'Win';
                  if(w > 0){
                      w++;
                  }else{
                        w = 1;
                  }
                  return null;
              }else{
                  output = ' ';
                  return null;
              }
          }else{
              if(input == 'rock'){
                  output = 'Win';
                  if(w > 0){
                        w++;
                  }else{
                      w = 1;
                  }
                  return null;
              }else if(input == 'paper'){
                  output = 'Lose';
                  if(l > 0){
                      l++;
                  }else{
                       l = 1;
                  }
                  return null;
              }else if(input == 'scissors'){
                  output = 'Tie';
                  if(t > 0){
                      t++;
                  }else{
                      t = 1;
                  }
                  return null;
              }else{
                  output = ' ';
                  return null;
              }
            }
            a.Finished__c = System.Today();
            UPDATE a;
        }
        
        userScore.Wins__c = w;
        userScore.Losses__c = l;
        userScore.Ties__c = t;
        userScore.Name = userInfo.getName();
        UPDATE userScore;
        
        available = null;
            return null;
    }
    
        //wrapper class for list
    public class aAsynch_RPS_Game {
        public Asynch_RPS_Game__c game {get; set;}
        public Boolean selected {get; set;}
        
        public aAsynch_RPS_Game(Asynch_RPS_Game__c g){
          game = g;
          selected = false;
        }
    }
}

Any help given is greatly appreciated!