The player is not always interested in a completely closed scenario. It is important that the player feels free to do what he/she feels necessary to do, even if it is not what you planned for them. Choice may allow the player to make an important decision that creates emotional investment, such as choosing to sacrifice a character or the main path he/she wants to take as their own. GUIDING the player is the best thing we could do, but it is totally different from FORCING the player to do something.
Example #1 (author that created the page): I will give as an example a game that has choices as its actual core mechanic: Sid Meier's Civilization, a game able to create a unique experience for the player, through giving the player the choice of nation, continents, strategies, etc.
Example #2: Another example is The Witcher series. The games focus a lot on storytelling and moral choices. Often they are made to shake the main characters (Geralt) own moral rules. He is a monster hunter and kills monsters as a job. But many times throughout the games, you will find humans that are acting like monsters and opposite. Most choices also have consequences. Not only choosing between who lives and dies, but also how people will treat you in the future. Doing this makes the player feel like it was his / hers choice, and you feel emotionally invested in it.
Example #3: Deus Ex: Mankind Divided. This game allows the player multiple ways to complete a mission objective. It doesn't force you to take a single path or do all of the side objectives that may be interesting story-wise but not necessary for the overall mission. This game also allows the player to choose different fighting/infiltration styles. If the player wants to go in guns blazing, then he is allowed to; if the player doesn't want to kill a single enemy throughout the entire game and still be able to beat the game, then he can do that.Example #4: Banjo Kazooie: Nuts & Bolts was designed around having multiple paths to victory. In this game players design and build vehicles to accomplish missions. The immense vehicle crafting mode allows for not just a few but countless different ways to approach missions. Players might want to approach a mission from the air with an aircraft, or stay on the ground with a monster truck that jumps over obstacles with springs.Example #5: Gravity Rush. In this game, the player is given a choice to update different stats and abilities to fit in her battle/gameplay style, such as increasing strength, recharge speed, or reducing resource cost. Also, the player can choose different ways to defeat enemies and traverse the map. This can not only give the player the ability to adjust the difficulty based on her experience, but also provides replay-ability by changing combat style.
Example 6: Star Wars: The Force Unleashed and Kingdoms of Amalur: Reckoning Edit
I have experienced this in the Star Wars: The Force Unleashed. In the game, there are events that present the choice to the player to either go on the dark path or continue Jedi way. Each path has their own consequence and benefits.
Second Game, which offers this mechanic is Kingdoms of Amalur: Reckoning, where the player has multiple choices in main quests, which in turn builds player's character and personality.