A more python efficient way to write a code

Instead of a complicated series of if statements, you could have an list or dictionary of (first, second) tuples,

a = [(1,3), (1,4), (2,1), (2,5) ...]

def give_winner(first_selection, second_selection):
    return (first_selection, second_selection) in a

You could also use a frozenset to improve performance.

You could use a dictionary.

dictionary = {
    1: [3, 4],
    2: [1, 5],
    3: [2, 4],
    4: [2, 5],
    5: [3, 1]
}

def give_winner(first_selection, second_selection):
    if dictionary.has_key(first_selection):
        if second_selection in dictionary[first_selection]:
            return True
    return False

You can make use of classes in python. For example, you can make player as a class with the following attributes:-

    Score
    Name
    OptionChosen

etc similarly, you can make methods like

    UpdateScore()
    DeclareWinner()

etc. This way your program will feel more "neat". You can also make a main() function which contains a

    while True:

and put all your content there. for ex

    class Player:
        def __init__(self,name, score = 0):
          self.name = name
          self.score = score  # initially score is zero
        def ChooseOption(self, name):
           if name == "computer":
            # select choice randomly code
           else:
             var = int(input("Enter choice: "))
        def UpdateScore(self):
            self.score += 1
    def main():
        player1 = Player("Name")
        player2 = Player("Computer") 
        while True:
         resp1 = player1.ChooseOption()
         resp2 = player2.ChooseOption()
         # add other code to manipulate resp1 and resp2 here 

and similarly you can code other things, Hope this gives you some idea

I enjoyed building my own little version of Rock Paper Scissor Lizard Spock.

You started your post by explaining som rules. So i figured, why not incorporate the rules inside the code. And i wanted to use real words instead of numbers since it would be easier to understand. But i agree it would be a hassle to spell scissors correct everytime, so i wanted numbers to work as input as well.

from random import randint

# ex. scissors and lizard is beaten by rock
beaten_by = {'rock': ['scissors', 'lizard'],
             'paper': ['rock', 'spock'],
             'scissors': ['paper', 'lizard'],
             'lizard': ['spock', 'paper'],
             'spock': ['scissors', 'rock']}

def rplsls_game():
    player_score, computer_score = 0, 0
    weapons = ['rock', 'paper', 'scissors', 'lizard', 'spock']

    while(abs(player_score - computer_score) < 2):

        print "-----------------------------------------------------------"
        print " Please enter your selection: "
        print " 1 (Rock), 2 (Paper), 3 (Scissors), 4 (Lizard) or 5 (Spock)"

        computer_weapon = weapons[randint(0, 4)]
        weapon = raw_input()
        if weapon in '12345': weapon = weapons[int(weapon) - 1]
        if weapon not in weapons:
            print "invalid input"
            continue        
        print "You selected: " + weapon
        print "Computer selected: " + computer_weapon

        if computer_weapon in beaten_by[weapon]:
            print "You won!"
            player_score += 1
        elif weapon in beaten_by[computer_weapon]:
            print "Computer won!"
            computer_score += 1
        else:
            print "Draw!"

        print "Player: " + str(player_score)
        print "Computer: " + str(computer_score)

    if player_score > computer_score:
        print "Congratulations! you won the game"
    else:
        print "Computer won the game..."

rplsls_game()

You can use also raw_input instand:

print("    Please enter your selection: 1 (Rock), 2 (Paper), 3 (Scissors), 4 (Lizard) or 5 (Spock): ")
selection = int(input())

try:
  selection = input("Please enter your selection: 1 (Rock), 2 (Paper), 3 (Scissors), 4 (Lizard) or 5 (Spock): ")
except ...

You complete forgot about exceptions.

And if stemtmants in your give_winner function is too large, use dictionar or lambda function.